{
  "info": {
    "id": "ses_f67cb8336ffe2mC4uaQqoou2Am",
    "slug": "glowing-circuit",
    "projectID": "global",
    "directory": "/private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-jPCAQg",
    "path": "private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-jPCAQg",
    "title": "Nudge content generation",
    "agent": "nudge-generation",
    "model": {
      "id": "gpt-6-astra",
      "providerID": "openai",
      "variant": "low"
    },
    "version": "1.18.30",
    "summary": {
      "additions": 0,
      "deletions": 0,
      "files": 0
    },
    "cost": 0,
    "tokens": {
      "input": 24979,
      "output": 3167,
      "reasoning": 191,
      "cache": {
        "read": 52992,
        "write": 0
      }
    },
    "time": {
      "created": 1789259971785,
      "updated": 1789260090867
    }
  },
  "messages": [
    {
      "info": {
        "role": "user",
        "time": {
          "created": 1789259971807
        },
        "agent": "nudge-generation",
        "model": {
          "providerID": "openai",
          "modelID": "gpt-6-astra",
          "variant": "low"
        },
        "system": "You create Codeforces learning content for Nudge. Your job is to understand the best approach for the problem and teach it clearly in your own words, using existing solutions to inform your reasoning.\n\nStart with a brief, targeted web search for existing solutions so relevant editorials are in context before substantial independent problem-solving. Use any supplied tutorial links first; otherwise search by contest ID and problem index. Prefer the official tutorial/editorial, and read relevant explanations rather than relying on search snippets. Look at accessible alternatives when useful, especially if the official explanation is missing, unclear, or unnecessarily complicated. Accepted submissions can also help clarify an approach. Respect supplied source lookup statuses and avoid repeatedly retrying blocked sources. If no usable source is accessible, proceed from the supplied statement.\n\nTreat existing solutions as a starting point, not a constraint. If an approach is clear and suitable, build on it without unnecessary re-derivation. If an explanation has gaps, seems incorrect, or uses a harder method than necessary, reason through it and derive a better or simpler approach when worthwhile. Choose a correct approach that is easy to teach and fits the constraints. Verify it against the supplied statement and edge cases before writing the hints, editorial, and C++ solution.\n\nThe editorial must be self-contained: explain the key insight, algorithm, why it works, complexity, and implementation details a learner needs, filling in steps the source skips. Keep research and verification concise; spend the output on teaching the solution, not narrating the research process. Follow the user prompt and output schema exactly.",
        "format": {
          "type": "json_schema",
          "schema": {
            "type": "object",
            "properties": {
              "status": {
                "type": "string",
                "enum": [
                  "success",
                  "unsolvable"
                ],
                "description": "Set to 'success' when the supplied statement defines a solvable programming problem. Set to 'unsolvable' only if the statement is fundamentally incomplete, contradictory, or depends on an inaccessible required resource."
              },
              "reason": {
                "anyOf": [
                  {
                    "type": "string",
                    "minLength": 1,
                    "maxLength": 2000
                  },
                  {
                    "type": "null"
                  }
                ],
                "description": "If status is 'unsolvable', provide a detailed failure message to show users. Name the concrete statement/resource blocker and any relevant source lookup/access status, such as missing official tutorial, 403/Cloudflare challenge, 404, or unavailable accepted submissions. Otherwise null."
              },
              "hints": {
                "anyOf": [
                  {
                    "type": "array",
                    "minItems": 5,
                    "maxItems": 5,
                    "description": "Exactly 5 progressive hints, each building on the last.",
                    "items": {
                      "type": "object",
                      "properties": {
                        "order": {
                          "type": "integer",
                          "minimum": 1,
                          "maximum": 5,
                          "description": "Hint number, 1 through 5."
                        },
                        "content": {
                          "type": "string",
                          "minLength": 1,
                          "maxLength": 10000,
                          "description": "Markdown hint text."
                        }
                      },
                      "required": [
                        "order",
                        "content"
                      ],
                      "additionalProperties": false
                    }
                  },
                  {
                    "type": "null"
                  }
                ],
                "description": "If status is 'success', provide exactly 5 progressive hints. Otherwise null."
              },
              "editorial": {
                "anyOf": [
                  {
                    "type": "string",
                    "minLength": 1,
                    "maxLength": 100000
                  },
                  {
                    "type": "null"
                  }
                ],
                "description": "If status is 'success', a prose editorial explaining the solution in Nudge's own voice, with no links, citations, or source/research notes. Otherwise null."
              },
              "solution": {
                "anyOf": [
                  {
                    "type": "string",
                    "minLength": 1,
                    "maxLength": 200000
                  },
                  {
                    "type": "null"
                  }
                ],
                "description": "If status is 'success', a complete C++ solution. Otherwise null."
              }
            },
            "required": [
              "status",
              "reason",
              "hints",
              "editorial",
              "solution"
            ],
            "additionalProperties": false,
            "title": "problem_response",
            "description": "Submit the generated content for the problem, or report if it is unsolvable."
          },
          "retryCount": 2
        },
        "id": "msg_098347cdf001kcKM6wljElijz5",
        "sessionID": "ses_f67cb8336ffe2mC4uaQqoou2Am"
      },
      "parts": [
        {
          "type": "text",
          "text": "Generate Codeforces learning content: five progressive hints, an original editorial, and AC-quality C++.\n\nFollow the editorial-first research workflow in the system instructions. Use the supplied statement to verify that the source solves this exact problem.\n\nWrite clean Markdown with LaTeX as needed. Use quick and clever humor when appropriate. Tell it like it is (don't sugar-coat responses), and use very casual language. You are fully allowed to swear, just don't overdo it like a sailor (be natural). Deconstruct any false assumptions.\n\nGenerate content for Codeforces problem 2260F: \"Edge Three-Coloring\".\n\nKnown problem metadata:\n- Contest ID: 2260\n- Index: F\n- Name: Edge Three-Coloring\n- Rating: unknown / unrated\n- Tags: bitmasks, brute force, dfs and similar, dsu, graphs, trees\n- Problem URL: https://codeforces.com/contest/2260/problem/F\n\nUse the rating and tags as weak signals only. The supplied statement is the source of truth.\n\nProblem Statement:\n<problem-statement>\n<div class=\"header\"><div class=\"title\">F. Edge Three-Coloring</div><div class=\"time-limit\"><div class=\"property-title\">time limit per test</div>4 seconds</div><div class=\"memory-limit\"><div class=\"property-title\">memory limit per test</div>512 megabytes</div><div class=\"input-file input-standard\"><div class=\"property-title\">input</div>standard input</div><div class=\"output-file output-standard\"><div class=\"property-title\">output</div>standard output</div></div><div><p>You are given an undirected connected graph with $$$n$$$ vertices and $$$m$$$ edges, where $$$m \\le n + 9$$$.</p><p>Each edge of the graph must be colored with one of three colors numbered from $$$1$$$ to $$$3$$$. We say that vertices $$$u$$$ and $$$v$$$ are reachable in color $$$c$$$ if there exists a path from $$$u$$$ to $$$v$$$ consisting only of edges of color $$$c$$$.</p><p>Determine whether there exists a way to color the edges such that the following conditions are satisfied:</p><ul> <li> for each color, there is at least one edge painted in that color; </li><li> for every pair of colors $$$i$$$ and $$$j$$$ and every pair of vertices $$$u$$$ and $$$v$$$, the following holds: if $$$u$$$ and $$$v$$$ are reachable in color $$$i$$$ and $$$i \\lt j$$$, then they are also reachable in color $$$j$$$. </li></ul></div><div class=\"input-specification\"><div class=\"section-title\">Input</div><p>The first line contains one integer $$$t$$$ ($$$1 \\le t \\le 1000$$$) — the number of test cases.</p><p>Each test case begins with a line that contains two integers $$$n$$$ and $$$m$$$ ($$$2 \\le n \\le 3000$$$; $$$n - 1 \\le m \\le n + 9$$$)&nbsp;— the number of vertices and edges in the graph.</p><p>Each of the next $$$m$$$ lines of a test case contains two integers $$$u_i$$$ and $$$v_i$$$ ($$$1 \\le u_i, v_i \\le n$$$, $$$u_i \\ne v_i$$$)&nbsp;— the endpoints of the $$$i$$$-th edge.</p><p>Additional constraints on the input:</p><ul> <li> the sum of $$$n$$$ over all test cases does not exceed $$$3000$$$; </li><li> in each test case, the graph has no self-loops or multiple edges; </li><li> in each test case, the graph is connected. </li></ul></div><div class=\"output-specification\"><div class=\"section-title\">Output</div><p>For each test case, print the answer as follows: if such an edge coloring exists, print <span class=\"tex-font-style-tt\">YES</span>; otherwise, print <span class=\"tex-font-style-tt\">NO</span>. Each letter may be printed in any case.</p></div><div class=\"sample-tests\"><div class=\"section-title\">Example</div><div class=\"sample-test\"><div class=\"input\"><div class=\"title\">Input</div><pre><div class=\"test-example-line test-example-line-even test-example-line-0\">4</div><div class=\"test-example-line test-example-line-odd test-example-line-1\">4 6</div><div class=\"test-example-line test-example-line-odd test-example-line-1\">1 2</div><div class=\"test-example-line test-example-line-odd test-example-line-1\">2 3</div><div class=\"test-example-line test-example-line-odd test-example-line-1\">3 1</div><div class=\"test-example-line test-example-line-odd test-example-line-1\">1 4</div><div class=\"test-example-line test-example-line-odd test-example-line-1\">2 4</div><div class=\"test-example-line test-example-line-odd test-example-line-1\">3 4</div><div class=\"test-example-line test-example-line-even test-example-line-2\">3 3</div><div class=\"test-example-line test-example-line-even test-example-line-2\">1 2</div><div class=\"test-example-line test-example-line-even test-example-line-2\">2 3</div><div class=\"test-example-line test-example-line-even test-example-line-2\">3 1</div><div class=\"test-example-line test-example-line-odd test-example-line-3\">7 9</div><div class=\"test-example-line test-example-line-odd test-example-line-3\">1 2</div><div class=\"test-example-line test-example-line-odd test-example-line-3\">2 3</div><div class=\"test-example-line test-example-line-odd test-example-line-3\">3 1</div><div class=\"test-example-line test-example-line-odd test-example-line-3\">1 4</div><div class=\"test-example-line test-example-line-odd test-example-line-3\">4 5</div><div class=\"test-example-line test-example-line-odd test-example-line-3\">5 1</div><div class=\"test-example-line test-example-line-odd test-example-line-3\">1 6</div><div class=\"test-example-line test-example-line-odd test-example-line-3\">6 7</div><div class=\"test-example-line test-example-line-odd test-example-line-3\">7 1</div><div class=\"test-example-line test-example-line-even test-example-line-4\">5 8</div><div class=\"test-example-line test-example-line-even test-example-line-4\">1 2</div><div class=\"test-example-line test-example-line-even test-example-line-4\">2 3</div><div class=\"test-example-line test-example-line-even test-example-line-4\">3 4</div><div class=\"test-example-line test-example-line-even test-example-line-4\">4 1</div><div class=\"test-example-line test-example-line-even test-example-line-4\">5 1</div><div class=\"test-example-line test-example-line-even test-example-line-4\">5 2</div><div class=\"test-example-line test-example-line-even test-example-line-4\">5 3</div><div class=\"test-example-line test-example-line-even test-example-line-4\">5 4</div></pre></div><div class=\"output\"><div class=\"title\">Output</div><pre><div class=\"test-example-line test-example-line-odd test-example-line-1\">YES</div><div class=\"test-example-line test-example-line-even test-example-line-2\">NO</div><div class=\"test-example-line test-example-line-odd test-example-line-3\">NO</div><div class=\"test-example-line test-example-line-even test-example-line-4\">YES</div></pre></div></div></div><div class=\"note\"><div class=\"section-title\">Note</div><p>In the first example, it's possible to paint the $$$5$$$-th edge in color $$$1$$$, the $$$1$$$-st and the $$$4$$$-th edge in color $$$2$$$, and all other edges in color $$$3$$$.</p></div>\n</problem-statement>\n\n\nSource lookup status:\n<source-lookup-status>\n- Tutorial: https://codeforces.com/blog/entry/156529 loaded (200 OK; title: \"Educational Codeforces Round 194 — Editorial - Codeforces\"; mentions 2260F)\n</source-lookup-status>\n\nGenerate:\n1. Five progressive hints, from a gentle nudge to the key insight.\n2. A self-contained editorial that teaches the solution clearly, including the key insight, algorithm, correctness argument, complexity, and necessary implementation details. Explain non-obvious steps without padding or a discovery-process transcript.\n3. A complete C++26 solution that gets AC on Codeforces.\n\nFormatting & Style Rules:\n- Hints and editorial: valid Markdown with LaTeX as needed (e.g., $dp[i]$, $$\\sum_{i=1}^{n} a_i$$).\n- Hints and editorial must read like Nudge's own explanation. No research notes, source notes, citations, Markdown links, URLs, or references to editorials/submissions/posts.\n- Solution: raw C++ only, no Markdown fences.\n- Keep C++ short, clean, standard, and single-file. Comments are fine when they clarify the idea.\n\nOutput strictness:\n- Return JSON matching the provided schema exactly.\n- If the problem is solvable, return `status: \"success\"`, `reason: null`, and fill in `hints`, `editorial`, and `solution`.\n- Return `status: \"unsolvable\"`, a short `reason`, and set `hints`, `editorial`, and `solution` to null only when the supplied statement is fundamentally incomplete, contradictory, or dependent on an inaccessible resource needed to define the task. Missing research, high rating, uncertainty, or \"can't guarantee AC\" is not a reason to return unsolvable.\n- If you return `status: \"unsolvable\"`, the `reason` is shown directly to users. Be concrete: say what exact statement/resource blocker stopped you, and include relevant source lookup status if it mattered (for example: official tutorial missing, Codeforces returned 403/Cloudflare challenge, tutorial page 404, accepted submissions unavailable).\n- Each hint must be JUST the hint text. No \"Hint 1:\" or subtitles. The UI adds those automatically.\n- Do not start the editorial with an \"# Editorial\" heading. The UI already adds that section. Feel free to add other headers as needed though.\n- Use styling in the editorial if needed! Just that first header is no good but any subheaders or bolding or whatever is good!\n\nFor the C++ solution, you MUST use this template and work around it:\n\n```cpp\n#include <bits/stdc++.h>\nusing namespace std;\n\nusing ll = long long;\n\nvoid setIO() {\n    ios::sync_with_stdio(false);\n    cin.tie(nullptr);\n}\n\nint main() {\n    setIO();\n}\n```",
          "id": "prt_098347ce0001Ng82d0s9aLGlE0",
          "sessionID": "ses_f67cb8336ffe2mC4uaQqoou2Am",
          "messageID": "msg_098347cdf001kcKM6wljElijz5"
        }
      ]
    },
    {
      "info": {
        "parentID": "msg_098347cdf001kcKM6wljElijz5",
        "role": "assistant",
        "mode": "nudge-generation",
        "agent": "nudge-generation",
        "variant": "low",
        "path": {
          "cwd": "/private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-jPCAQg",
          "root": "/"
        },
        "cost": 0,
        "tokens": {
          "total": 9134,
          "input": 9102,
          "output": 32,
          "reasoning": 0,
          "cache": {
            "write": 0,
            "read": 0
          }
        },
        "modelID": "gpt-6-astra",
        "providerID": "openai",
        "time": {
          "created": 1789259971939,
          "completed": 1789259977861
        },
        "finish": "tool-calls",
        "id": "msg_098347d63001Qcxi9vSfoVSCeq",
        "sessionID": "ses_f67cb8336ffe2mC4uaQqoou2Am"
      },
      "parts": [
        {
          "type": "step-start",
          "id": "prt_098348b600010ozlW32lJzaQ7p",
          "sessionID": "ses_f67cb8336ffe2mC4uaQqoou2Am",
          "messageID": "msg_098347d63001Qcxi9vSfoVSCeq"
        },
        {
          "type": "tool",
          "tool": "webfetch",
          "callID": "call_hKrpdnSauN5KSoOeo5RGv2TZ",
          "state": {
            "status": "completed",
            "input": {
              "url": "https://codeforces.com/blog/entry/156529",
              "format": "markdown"
            },
            "output": "---\nimage: https://codeforces.com/codeforces.org/s/13758/images/codeforces-telegram-square-1024x1024.png\ntitle: Educational Codeforces Round 194 — Editorial - Codeforces\ndescription: Codeforces. Programming competitions and contests, programming community\n---\n\n<!DOCTYPE HTML PUBLIC \"-//W3C//DTD HTML 4.01//EN\"> \n\n  \n[![Codeforces](//codeforces.com/codeforces.org/s/13758/images/codeforces-logo-with-telegram-555x130.png \"Codeforces\")](https://codeforces.com/) \n\n[![In English](//codeforces.com/codeforces.org/s/13758/images/flags/24/gb.png \"In English\")](https://codeforces.com/blog/entry/156529?locale=en) [![По-русски](//codeforces.com/codeforces.org/s/13758/images/flags/24/ru.png \"По-русски\")](https://codeforces.com/blog/entry/156529?locale=ru) \n\n[Enter](https://codeforces.com/enter?back=%2Fblog%2Fentry%2F156529)| [Register](https://codeforces.com/register) \n\n  \n* [Home](https://codeforces.com/)\n* [Top](https://codeforces.com/top)\n* [Catalog](https://codeforces.com/catalog)\n* [Contests](https://codeforces.com/contests)\n* [Gym](https://codeforces.com/gyms)\n* [Problemset](https://codeforces.com/problemset)\n* [Groups](https://codeforces.com/groups)\n* [Rating](https://codeforces.com/ratings)\n* [Edu](https://codeforces.com/edu/courses)\n* [API](https://codeforces.com/apiHelp)\n* [Calendar](https://codeforces.com/calendar)\n* [Help](https://codeforces.com/help)\n  \n  \n→ Pay attention \n\nBefore contest  \n[Codeforces Round 1121 (Div. 2)](https://codeforces.com/contests/2264)  \n16:25:24  \n[Register now »](https://codeforces.com/contestRegistration/2264)\n\n\\*has extra registration\n\n→ Streams \n\n[ICPC India Regionals 2026 QnA](https://codeforces.com/stream/1043) \n\nBy [aryanc403](https://codeforces.com/profile/aryanc403 \"Master aryanc403\") \n\nBefore stream 38:55:24 \n\n|  | [View all →](https://codeforces.com/streams) |\n|  | -------------------------------------------- |\n\n→ Top rated \n\n| #  | User                                                                                          | Rating |\n| -- | --------------------------------------------------------------------------------------------- | ------ |\n| 1  | [jiangly](https://codeforces.com/profile/jiangly \"Legendary Grandmaster jiangly\")             | 3810   |\n| 2  | [Kevin114514](https://codeforces.com/profile/Kevin114514 \"Legendary Grandmaster Kevin114514\") | 3655   |\n| 3  | [Benq](https://codeforces.com/profile/Benq \"Legendary Grandmaster Benq\")                      | 3650   |\n| 4  | [maroonrk](https://codeforces.com/profile/maroonrk \"Legendary Grandmaster maroonrk\")          | 3427   |\n| 5  | [strapple](https://codeforces.com/profile/strapple \"Legendary Grandmaster strapple\")          | 3413   |\n| 6  | [Um\\_nik](https://codeforces.com/profile/Um%5Fnik \"Legendary Grandmaster Um_nik\")             | 3387   |\n| 7  | [heuristica](https://codeforces.com/profile/heuristica \"Legendary Grandmaster heuristica\")    | 3322   |\n| 8  | [tourist](https://codeforces.com/profile/tourist \"Legendary Grandmaster tourist\")             | 3301   |\n| 9  | [turmax](https://codeforces.com/profile/turmax \"Legendary Grandmaster turmax\")                | 3297   |\n| 10 | [squareOf105](https://codeforces.com/profile/squareOf105 \"Legendary Grandmaster squareOf105\") | 3262   |\n\n| [Countries](https://codeforces.com/ratings/countries) \\| [Cities](https://codeforces.com/ratings/cities) | [Organizations](https://codeforces.com/ratings/organizations) | [View all →](https://codeforces.com/ratings) |\n| -------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------- | -------------------------------------------- |\n\n→ Top contributors \n\n| #  | User                                                                                     | Contrib. |\n| -- | ---------------------------------------------------------------------------------------- | -------- |\n| 1  | [Qingyu](https://codeforces.com/profile/Qingyu \"Legendary Grandmaster Qingyu\")           | 157      |\n| 2  | [maspy](https://codeforces.com/profile/maspy \"Legendary Grandmaster maspy\")              | 150      |\n| 3  | [nik\\_exists](https://codeforces.com/profile/nik%5Fexists \"Expert nik_exists\")           | 149      |\n| 4  | [Um\\_nik](https://codeforces.com/profile/Um%5Fnik \"Legendary Grandmaster Um_nik\")        | 145      |\n| 5  | [Errichto](https://codeforces.com/profile/Errichto \"International Grandmaster Errichto\") | 139      |\n| 6  | [adamant](https://codeforces.com/profile/adamant \"Master adamant\")                       | 136      |\n| 7  | [maroonrk](https://codeforces.com/profile/maroonrk \"Legendary Grandmaster maroonrk\")     | 134      |\n| 8  | [AmShZ](https://codeforces.com/profile/AmShZ \"Grandmaster AmShZ\")                        | 133      |\n| 8  | [BledDest](https://codeforces.com/profile/BledDest \"International Grandmaster BledDest\") | 133      |\n| 10 | [DNR](https://codeforces.com/profile/DNR \"Master DNR\")                                   | 131      |\n\n|  | [View all →](https://codeforces.com/top-contributed) |\n|  | ---------------------------------------------------- |\n\n→ Find user \n\nHandle: \n\n→ Recent actions \n\n* [kondasujay2](https://codeforces.com/profile/kondasujay2 \"International Master kondasujay2\") → [Codeforces Round 1120 (Div 1, Div 2) Editorial](https://codeforces.com/blog/entry/156688) ![New comment\\(s\\)](//codeforces.com/codeforces.org/s/13758/images/icons/comment-12x12.png \"New comment(s)\")\n* [grecil](https://codeforces.com/profile/grecil \"Expert grecil\") → [Funny Story](https://codeforces.com/blog/entry/156682) ![New comment\\(s\\)](//codeforces.com/codeforces.org/s/13758/images/icons/comment-12x12.png \"New comment(s)\")\n* [kondasujay2](https://codeforces.com/profile/kondasujay2 \"International Master kondasujay2\") → [Codeforces Round 1120 (Div. 1, Div. 2)](https://codeforces.com/blog/entry/156576) ![New comment\\(s\\)](//codeforces.com/codeforces.org/s/13758/images/icons/comment-12x12.png \"New comment(s)\")\n* [qwexd](https://codeforces.com/profile/qwexd \"Master qwexd\") → [Codeforces Round 1121 (Div. 2)](https://codeforces.com/blog/entry/156653) ![New comment\\(s\\)](//codeforces.com/codeforces.org/s/13758/images/icons/comment-12x12.png \"New comment(s)\")\n* [BledDest](https://codeforces.com/profile/BledDest \"International Grandmaster BledDest\") → [Educational Codeforces Round 194 - Editorial](https://codeforces.com/blog/entry/156529) ![New comment\\(s\\)](//codeforces.com/codeforces.org/s/13758/images/icons/comment-12x12.png \"New comment(s)\")\n* [sugim48](https://codeforces.com/profile/sugim48 \"Grandmaster sugim48\") → [Spaghetti Network — a graph puzzle game](https://codeforces.com/blog/entry/156563) ![New comment\\(s\\)](//codeforces.com/codeforces.org/s/13758/images/icons/comment-12x12.png \"New comment(s)\")\n* [MikeMirzayanov](https://codeforces.com/profile/MikeMirzayanov \"Headquarters, MikeMirzayanov\") → [Telegram Returns as Title Sponsor of Codeforces!](https://codeforces.com/blog/entry/156620) ![New comment\\(s\\)](//codeforces.com/codeforces.org/s/13758/images/icons/comment-12x12.png \"New comment(s)\")\n* [ismailfateen](https://codeforces.com/profile/ismailfateen \"Expert ismailfateen\") → [ACPC 2026 Qualified Teams](https://codeforces.com/blog/entry/156227) ![Text created or updated](//codeforces.com/codeforces.org/s/13758/images/icons/x-update-12x12.png \"Text created or updated\")\n* [saksham2005mittal](https://codeforces.com/profile/saksham2005mittal \"Newbie saksham2005mittal\") → [Codeforces Round 1120 (Div. 2) — B — Min Matrices](https://codeforces.com/blog/entry/156686) ![Text created or updated](//codeforces.com/codeforces.org/s/13758/images/icons/x-update-12x12.png \"Text created or updated\")\n* [CrazyCoder00](https://codeforces.com/profile/CrazyCoder00 \"Specialist CrazyCoder00\") → [Visualize your practice vs rating growth: Codeforces Peak Analyzer (Browser Extension)](https://codeforces.com/blog/entry/156687) ![New comment\\(s\\)](//codeforces.com/codeforces.org/s/13758/images/icons/comment-12x12.png \"New comment(s)\")\n* [turmax](https://codeforces.com/profile/turmax \"Legendary Grandmaster turmax\") → [Are we ready for AGI?](https://codeforces.com/blog/entry/156668) ![New comment\\(s\\)](//codeforces.com/codeforces.org/s/13758/images/icons/comment-12x12.png \"New comment(s)\")\n* [mariza\\_CY](https://codeforces.com/profile/mariza%5FCY \"Candidate Master mariza_CY\") → [Balkan OI 2026 teams](https://codeforces.com/blog/entry/153535) ![New comment\\(s\\)](//codeforces.com/codeforces.org/s/13758/images/icons/comment-12x12.png \"New comment(s)\")\n* [nik\\_exists](https://codeforces.com/profile/nik%5Fexists \"Expert nik_exists\") → [Codeforces Round 1119 (Div. 3) Editorial](https://codeforces.com/blog/entry/156457) ![New comment\\(s\\)](//codeforces.com/codeforces.org/s/13758/images/icons/comment-12x12.png \"New comment(s)\")\n* [Eason\\_He](https://codeforces.com/profile/Eason%5FHe \"Master Eason_He\") → [2026 Kruskal Cup](https://codeforces.com/blog/entry/156137) ![New comment\\(s\\)](//codeforces.com/codeforces.org/s/13758/images/icons/comment-12x12.png \"New comment(s)\")\n* [aryanc403](https://codeforces.com/profile/aryanc403 \"Master aryanc403\") → [ICPC India Regionals 2026 QnA](https://codeforces.com/blog/entry/156683) ![Text created or updated](//codeforces.com/codeforces.org/s/13758/images/icons/x-update-12x12.png \"Text created or updated\")\n* [BledDest](https://codeforces.com/profile/BledDest \"International Grandmaster BledDest\") → [Educational Codeforces Round 189 — Editorial](https://codeforces.com/blog/entry/153154) ![Text created or updated](//codeforces.com/codeforces.org/s/13758/images/icons/x-update-12x12.png \"Text created or updated\")\n* [atcoder\\_official](https://codeforces.com/profile/atcoder%5Fofficial \"Unrated, atcoder_official\") → [AtCoder Beginner Contest 475 Announcement](https://codeforces.com/blog/entry/156627) ![New comment\\(s\\)](//codeforces.com/codeforces.org/s/13758/images/icons/comment-12x12.png \"New comment(s)\")\n* [hofung614](https://codeforces.com/profile/hofung614 \"Newbie hofung614\") → [Is Problem 20B the most over-rated problem on Codeforces?](https://codeforces.com/blog/entry/156663) ![New comment\\(s\\)](//codeforces.com/codeforces.org/s/13758/images/icons/comment-12x12.png \"New comment(s)\")\n* [atcoder\\_official](https://codeforces.com/profile/atcoder%5Fofficial \"Unrated, atcoder_official\") → [AtCoder Heuristic Contest 071 Announcement](https://codeforces.com/blog/entry/156676) ![Text created or updated](//codeforces.com/codeforces.org/s/13758/images/icons/x-update-12x12.png \"Text created or updated\")\n* [p0uya3767](https://codeforces.com/profile/p0uya3767 \"Newbie p0uya3767\") → [Carot is broke ](https://codeforces.com/blog/entry/141823) ![Text created or updated](//codeforces.com/codeforces.org/s/13758/images/icons/x-update-12x12.png \"Text created or updated\") ![Necropost](//codeforces.com/codeforces.org/s/13758/images/icons/hourglass.png \"Necropost\")\n* [Arbuzik342](https://codeforces.com/profile/Arbuzik342 \"Candidate Master Arbuzik342\") → [Best April Fools problems in my opinion](https://codeforces.com/blog/entry/156674) ![New comment\\(s\\)](//codeforces.com/codeforces.org/s/13758/images/icons/comment-12x12.png \"New comment(s)\")\n* [pranavdhardwivedi](https://codeforces.com/profile/pranavdhardwivedi \"Pupil pranavdhardwivedi\") → [CodeRush 3.0 - Ready to Race Against the Best ?](https://codeforces.com/blog/entry/156343) ![New comment\\(s\\)](//codeforces.com/codeforces.org/s/13758/images/icons/comment-12x12.png \"New comment(s)\")\n* [brunomont](https://codeforces.com/profile/brunomont \"Master brunomont\") → [tgen::miniblog(4): random trees](https://codeforces.com/blog/entry/156658) ![New comment\\(s\\)](//codeforces.com/codeforces.org/s/13758/images/icons/comment-12x12.png \"New comment(s)\")\n* [MrPupsik](https://codeforces.com/profile/MrPupsik \"Expert MrPupsik\") → [orzchat update](https://codeforces.com/blog/entry/156617) ![New comment\\(s\\)](//codeforces.com/codeforces.org/s/13758/images/icons/comment-12x12.png \"New comment(s)\")\n* [mrbluevn](https://codeforces.com/profile/mrbluevn \"Specialist mrbluevn\") → [I'm going to hit a 100-day Codeforces streak!](https://codeforces.com/blog/entry/156664) ![New comment\\(s\\)](//codeforces.com/codeforces.org/s/13758/images/icons/comment-12x12.png \"New comment(s)\")\n\n|  | [Detailed →](https://codeforces.com/recent-actions) |\n|  | --------------------------------------------------- |\n\n* [BledDest](https://codeforces.com/profile/BledDest)\n* [Blog](https://codeforces.com/blog/BledDest)\n* [Teams](https://codeforces.com/teams/with/BledDest)\n* [Submissions](https://codeforces.com/submissions/BledDest)\n* [Groups](https://codeforces.com/groups/with/BledDest)\n* [Contests](https://codeforces.com/contests/with/BledDest)\n* [Problemsetting](https://codeforces.com/contests/writer/BledDest)\n\n### [BledDest's blog](https://codeforces.com/blog/BledDest)\n\n[ Educational Codeforces Round 194 — Editorial ](https://codeforces.com/blog/entry/156529) \n\nBy [BledDest](https://codeforces.com/profile/BledDest \"International Grandmaster BledDest\"), 4 days ago, translation, ![In English](//codeforces.com/codeforces.org/s/13758/images/flags/24/gb.png \"In English\") \n\n[2260A - Monocarp's Contest](https://codeforces.com/contest/2260/problem/A \"Educational Codeforces Round 194 (Rated for Div. 2)\")\n\n**Tutorial**\n\nTutorial is loading...\n\n**Solution**\n\n```\nt = int(input())\nfor i in range(t):\n    n = int(input())\n    a = list(map(int, input().split()))\n    \n    cnt0 = a.count(0)\n    if cnt0 >= 2:\n        print(a[0] + a[-1])\n    else:\n        print(-1)\n\n\n```\n\n[2260B - Monocarp and Projects](https://codeforces.com/contest/2260/problem/B \"Educational Codeforces Round 194 (Rated for Div. 2)\")\n\n**Tutorial**\n\nTutorial is loading...\n\n**Solution**\n\n```\n#include<bits/stdc++.h>\n\nusing namespace std;\n\nvoid solve()\n{\n    long long x, y, k;\n    cin >> x >> y >> k;\n    long long ans = 0;\n    for(long long i = 0; i < min(y, k); i++)\n        ans += (y + i) % (x + i);\n    long long full = max(0ll, k - y);\n    ans += full * (y - x);\n    cout << ans << endl;    \n}\n\nint main()\n{\n    ios_base::sync_with_stdio(0);\n    cin.tie(0);\n    int t;\n    cin >> t;\n    for(int i = 0; i < t; i++)\n        solve();\n}\n\n\n```\n\n[2260C - Maximize XOR, Minimize Operations](https://codeforces.com/contest/2260/problem/C \"Educational Codeforces Round 194 (Rated for Div. 2)\")\n\n**Tutorial**\n\nTutorial is loading...\n\n**Solution**\n\n```\n#include<bits/stdc++.h>\n\nusing namespace std;\n\nvoid solve()\n{\n    int x, y;\n    cin >> x >> y;\n    int s = x + y;\n    for(int d = (1 << 30); d >= 1; d >>= 1)\n        if((s & d) != 0 && x >= d)\n            x -= d;\n    cout << s << \" \" << x << endl;    \n}\n\nint main()\n{\n    ios_base::sync_with_stdio(0);\n    cin.tie(0);\n    int t;\n    cin >> t;\n    for(int i = 0; i < t; i++)\n        solve();\n}\n\n\n```\n\n[2260D - Signs of Prefix Sums](https://codeforces.com/contest/2260/problem/D \"Educational Codeforces Round 194 (Rated for Div. 2)\")\n\n**Tutorial**\n\nTutorial is loading...\n\n**Solution**\n\n```\n#include<bits/stdc++.h>\n\nusing namespace std;\n\nconst int K = 3;\nconst int INF = int(1e9);\n\npair<int, int> get_segment(char c)\n{\n    if(c == '0') return {K, K};\n    if(c == '-') return {0, K - 1};\n    return {K + 1, K * 2};\n}\n\nvoid solve()\n{\n    int n;\n    string s;\n    cin >> n >> s;\n    vector<vector<int>> dp(n + 1, vector<int>(2 * K + 1, INF));\n    dp[0][K] = 0;\n    for(int i = 0; i < n; i++)\n        for(int j = 0; j < 2 * K + 1; j++)\n        {\n            if(dp[i][j] == INF) continue;\n            auto [l, r] = get_segment(s[i]);\n            for(int k = l; k <= r; k++)\n            {\n                if(k == j) continue;\n                int& d = dp[i + 1][k];\n                d = min(d, max(dp[i][j], abs(k - j)));\n            }\n        }\n    int ans = *min_element(dp[n].begin(), dp[n].end());\n    if(ans == INF) ans = -1;\n    cout << ans << endl;\n}\n\nint main()\n{\n    ios_base::sync_with_stdio(0);\n    cin.tie(0);\n    int t;\n    cin >> t;\n    for(int i = 0; i < t; i++)\n        solve();    \n}\n\n\n```\n\n[2260E - Cyclic Balance](https://codeforces.com/contest/2260/problem/E \"Educational Codeforces Round 194 (Rated for Div. 2)\")\n\n**Tutorial**\n\nTutorial is loading...\n\n**Solution**\n\n```\n#include<bits/stdc++.h>\n\nusing namespace std;\n\nint get(int c00, int c11, int cd)\n{                                      \n    int len = c00 + c11 + cd;\n    int lf = len / 4;\n    int rg = 1e8;\n    int ans = 1e8;\n    while(rg >= lf)\n    {\n        int mid = (lf + rg) / 2;\n        int need_break = max(0, c00 - mid) + max(0, c11 - mid);\n        if(need_break + cd / 2 <= mid)\n        {\n            ans = mid;\n            rg = mid - 1;\n        }\n        else\n            lf = mid + 1;\n    }\n    return ans;\n}\n\nvoid solve()\n{\n    int n, q;\n    cin >> n >> q;\n    string s;\n    cin >> s;\n    vector<int> pref1(n + 1), prefdiff(n);\n    for(int i = 0; i < n; i++)\n        pref1[i + 1] = pref1[i] + (s[i] - '0');\n    for(int i = 0; i + 1 < n; i++)\n        prefdiff[i + 1] = prefdiff[i] + (s[i] != s[i + 1]);\n\n    for(int i = 0; i < q; i++)\n    {\n        int l, r;\n        cin >> l >> r;\n        if(l == r)\n            cout << 3 << \"\\n\";\n        else\n        {\n            int c1 = pref1[r] - pref1[l - 1];\n            int cd = prefdiff[r - 1] - prefdiff[l - 1];\n            if(s[l - 1] != s[r - 1])\n                cd++;\n            int c0 = (r - l + 1) - c1;\n            int c00 = c0 - (cd / 2);\n            int c11 = c1 - (cd / 2);\n            cout << get(c00, c11, cd) * 4 - (r - l + 1) << \"\\n\";\n        }\n    }\n}\n\nint main()\n{\n    ios_base::sync_with_stdio(0);\n    cin.tie(0);\n    int t = 1;\n    //cin >> t;\n    for(int i = 0; i < t; i++)\n        solve();\n}\n\n\n```\n\n[2260F - Edge Three-Coloring](https://codeforces.com/contest/2260/problem/F \"Educational Codeforces Round 194 (Rated for Div. 2)\")\n\n**Tutorial**\n\nTutorial is loading...\n\n**Solution**\n\n```\n#include<bits/stdc++.h>\n\nusing namespace std;\n\nconst int N = 3010;\n\nvector<pair<int, int>> g[N];\nint p[N];\nint pe[N];\nint used[N];\nint n, m;\nvector<int> bad_edges;\nint cycle[N];\nint xs[N], ys[N];\n\nvoid dfs1(int x)\n{\n    for(auto [y, i] : g[x])\n        if(p[y] == -1)\n        {\n            p[y] = x;\n            pe[y] = i;\n            dfs1(y);\n        }\n        else if(p[x] != y)\n            bad_edges.push_back(i);\n}\n\nvoid go_to_root(int x)\n{\n    while(x != 0)\n    {\n        cycle[pe[x]] ^= 1;\n        x = p[x];\n    }\n}\n\nvoid dfs2(int x)\n{\n    used[x] = true;\n    for(auto [y, i] : g[x])\n        if(cycle[i] == 0 && !used[y])\n            dfs2(y); \n}\n\nvoid solve()\n{\n    cin >> n >> m;\n    bad_edges.clear();\n    for(int i = 0; i < n; i++)\n    {\n        g[i].clear();\n        p[i] = -1;\n        pe[i] = -1;\n    }\n    for(int i = 0; i < m; i++)\n    {\n        int x, y;\n        cin >> x >> y;\n        --x;\n        --y;\n        xs[i] = x;\n        ys[i] = y;\n        g[x].push_back(make_pair(y, i));\n        g[y].push_back(make_pair(x, i));\n    }\n    \n    p[0] = 0;\n    dfs1(0);\n    sort(bad_edges.begin(), bad_edges.end());\n    bad_edges.erase(unique(bad_edges.begin(), bad_edges.end()), bad_edges.end());\n\n    int k = bad_edges.size();\n    for(int mask = 1; mask < (1 << k); mask++)\n    {\n        for(int edge = 0; edge < m; edge++) cycle[edge] = 0;\n        for(int bit = 0; bit < k; bit++)\n            if((mask >> bit) & 1)\n            {                                                               \n                go_to_root(xs[bad_edges[bit]]);\n                go_to_root(ys[bad_edges[bit]]);\n                cycle[bad_edges[bit]] ^= 1;\n            }\n        for(int vertex = 0; vertex < n; vertex++)\n            used[vertex] = 0;\n        dfs2(0);\n        bool good = true;\n        for(int vertex = 0; vertex < n; vertex++)\n            if(!used[vertex])\n                good = false;\n        if(good)\n        {\n            cout << \"YES\\n\";\n            return;\n        }\n    }\n    cout << \"NO\\n\";\n}\n\nint main()\n{\n    ios_base::sync_with_stdio(0);\n    cin.tie(0);\n    int t;\n    cin >> t;\n    for(int i = 0; i < t; i++)\n        solve();    \n}\n\n\n```\n\n[2260G - Sortable Permutations](https://codeforces.com/contest/2260/problem/G \"Educational Codeforces Round 194 (Rated for Div. 2)\")\n\n**Tutorial**\n\nTutorial is loading...\n\n**Solution**\n\n```\n#include<bits/stdc++.h>\n\nusing namespace std;\n\nconst int MOD = 998244353;\n\nint add(int x, int y)\n{\n    x += y;\n    while(x >= MOD) x -= MOD;\n    while(x < 0) x += MOD;\n    return x;   \n}\n\nint mul(int x, int y)\n{\n    return (x * 1ll * y) % MOD;\n}\n\nint binpow(int x, int y)\n{\n    int z = 1;\n    while(y > 0)\n    {\n        if(y % 2 == 1) z = mul(z, x);\n        x = mul(x, x);\n        y /= 2;\n    }\n    return z;\n}\n\nint inv(int x)\n{\n    return binpow(x, MOD - 2);\n}\n\nconst int N = 200043;\n\nint mind[N];\nint fact[N];\nint ifact[N];\n\nvoid precalc()\n{\n    for(int i = 2; i < N; i++)\n        for(int j = i; j < N; j += i)\n            if(mind[j] == 0)\n                mind[j] = i;\n\n    fact[0] = 1;\n    for(int i = 1; i < N; i++)\n        fact[i] = mul(fact[i - 1], i);\n    ifact[N - 1] = inv(fact[N - 1]);\n    for(int i = N - 1; i >= 1; i--)\n        ifact[i - 1] = mul(ifact[i], i);\n}\n\nint choose(int n, int k)\n{\n    if(n < 0 || n < k || k < 0) return 0;\n    return mul(fact[n], mul(ifact[k], ifact[n - k]));\n}\n\nint n;\n\nvector<int> factorize(int x)\n{\n    vector<int> res;\n    while(x != 1)\n    {\n        res.push_back(mind[x]);\n        x /= mind[x];\n    }\n    return res;\n}\n\nvoid gen(const vector<int>& d, int i, long long cur, int m, vector<pair<long long, int>>& res)\n{\n    if(i == d.size()) res.push_back(make_pair(cur, m));\n    else\n    {\n        gen(d, i + 1, cur, m, res);\n        gen(d, i + 1, cur * 1ll * d[i], -m, res);\n    }\n} \n\nvoid solve()\n{\n    cin >> n;\n    vector<pair<pair<long long, int>, int>> edges;\n    for(int i = 1; i < n; i++)\n    {\n        auto f1 = factorize(i);\n        auto f2 = factorize(i + 1);\n        for(auto x : f2) f1.push_back(x);\n        sort(f1.begin(), f1.end());\n        f1.erase(unique(f1.begin(), f1.end()), f1.end());\n        vector<pair<long long, int>> pos;\n        gen(f1, 0, 1, 1, pos);\n        for(auto [x, s] : pos)\n        {\n            if(x == 1) continue;\n            edges.push_back(make_pair(make_pair(x, s), i));\n        }\n    }\n    \n    int ans = 1;\n    int l = 0;\n    sort(edges.begin(), edges.end());\n    while(l < edges.size())\n    {\n        int r = l;\n        while(r < edges.size() && edges[r].first.first == edges[l].first.first)\n            r++;\n        vector<int> e;\n        for(int i = l; i < r; i++)\n            e.push_back(edges[i].second);\n        long long i = edges[l].first.first;\n        int m = edges[l].first.second;\n        if(m != 0)\n        {\n            int absent = n / i;\n            int cur = mul(fact[n], ifact[n - absent]);\n            int lf = 0;\n            while(lf < e.size())\n            {\n                int rg = lf;\n                int nxt = e[lf];\n                while(rg < e.size() && e[rg] == nxt)\n                {\n                    rg++;\n                    nxt++;\n                }\n                vector<int> cnt(2);\n                for(int k = e[lf]; k <= e[rg - 1] + 1; k++)\n                {\n                    if(k % i != 0)\n                        cnt[k % 2]++;     \n                }\n                //cout << cnt[0] << \" \" << cnt[1] << endl;\n                cur = mul(cur, choose(cnt[0] + cnt[1], cnt[0]));\n                lf = rg;\n            }\n            //cout << i << \" \" << cur << endl;\n            ans = add(ans, -1 * m * add(cur, -1));\n        }\n        l = r;\n    }\n    cout << ans << endl;\n}\n\nint main()\n{\n    ios_base::sync_with_stdio(0);\n    cin.tie(0);\n    precalc();\n    int t = 1;\n    //cin >> t;\n    for(int i = 0; i < t; i++)\n        solve();\n}\n\n```\n\n![](//codeforces.com/codeforces.org/s/13758/images/icons/paperclip-16x16.png) Tutorial of [Educational Codeforces Round 194 (Rated for Div. 2)](https://codeforces.com/contest/2260) \n\n* [![Vote: I like it](//codeforces.com/codeforces.org/s/13758/images/actions/voteup-gray.png \"Vote: I like it\")](#)\n* +91\n* [![Vote: I do not like it](//codeforces.com/codeforces.org/s/13758/images/actions/votedown-gray.png \"Vote: I do not like it\")](#)\n\n* [![Author](//codeforces.com/codeforces.org/s/13758/images/blog/user_16x16.png \"Author\")](https://codeforces.com/profile/BledDest) [BledDest ](https://codeforces.com/profile/BledDest)\n* ![Publication date](//codeforces.com/codeforces.org/s/13758/images/blog/date_16x16.png \"Publication date\") 4 days ago\n* [![Comments](//codeforces.com/codeforces.org/s/13758/images/blog/comments_16x16.png \"Comments\")](https://codeforces.com/blog/entry/156529#comments) [41 ](https://codeforces.com/blog/entry/156529#comments)\n\n  \n![Comments](//codeforces.com/codeforces.org/s/13758/images/icons/comments-48x48.png \"Comments\") Comments (41) \n\n[Write comment?](#) \n\n| » [BledDest](https://codeforces.com/profile/BledDest \"International Grandmaster BledDest\") | 4 days ago, [show (+1)](#) # |\n| ------------------------------------------------------------------------------------------ | ---------------------------- |\n\n| » [ ![](https://codeforces.com/userpic.codeforces.org/331814/avatar/36406af67e5558a6.jpg) ](https://codeforces.com/profile/BledDest) [BledDest](https://codeforces.com/profile/BledDest \"International Grandmaster BledDest\") | 4 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156529?#comment-1391195 \"Link to comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) The tutorials for problems will be available in a few minutes. [→](#) [Reply](#) |\n| ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n* | » » [Askew\\_breadcrumb6](https://codeforces.com/profile/Askew%5Fbreadcrumb6 \"Newbie Askew_breadcrumb6\") | 2 days ago, [show (+1)](#) # |\n| ------------------------------------------------------------------------------------------------------- | ---------------------------- |\n\n| » » [ ![](https://codeforces.com/userpic.codeforces.org/2572984/avatar/6e314e22b534732f.jpg) ](https://codeforces.com/profile/Askew%5Fbreadcrumb6) [Askew\\_breadcrumb6](https://codeforces.com/profile/Askew%5Fbreadcrumb6 \"Newbie Askew_breadcrumb6\") | 2 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156529?#comment-1391575 \"Link to comment\") [^](#comment-1391195 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) In problem C, what's preventing us from having a greedy pick where, if in the sum S=X+Y there is a 0 bit, it is a result of 1 in X and 1 in Y?In other words, Why can't I have 1 ^ 1 resulting in 0 in S?Why should I be mapping the set bits of S to X and Y? Why not have a configuration where unset bits are a result of 1 bit in X and 1 bit in Y? [→](#) [Reply](#) |\n| ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ |\n\n  * | » » » [amey75](https://codeforces.com/profile/amey75 \"Specialist amey75\") | 40 hours ago, [show (+1)](#) # |\n| ------------------------------------------------------------------------- | ------------------------------ |\n\n| » » » [ ![](https://codeforces.com/userpic.codeforces.org/4580660/avatar/53b3970dc02dcbd7.jpg) ](https://codeforces.com/profile/amey75) [amey75](https://codeforces.com/profile/amey75 \"Specialist amey75\") | 40 hours ago, [hide](#) [#](https://codeforces.com/blog/entry/156529?#comment-1391662 \"Link to comment\") [^](#comment-1391575 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) cause that would lead to X&Y being non zero. [→](#) [Reply](#) |\n| ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n    * | » » » » [Askew\\_breadcrumb6](https://codeforces.com/profile/Askew%5Fbreadcrumb6 \"Newbie Askew_breadcrumb6\") | 39 hours ago, [show](#) # |\n| ----------------------------------------------------------------------------------------------------------- | ------------------------- |\n\n| » » » » [ ![](https://codeforces.com/userpic.codeforces.org/2572984/avatar/6e314e22b534732f.jpg) ](https://codeforces.com/profile/Askew%5Fbreadcrumb6) [Askew\\_breadcrumb6](https://codeforces.com/profile/Askew%5Fbreadcrumb6 \"Newbie Askew_breadcrumb6\") | 39 hours ago, [hide](#) [#](https://codeforces.com/blog/entry/156529?#comment-1391668 \"Link to comment\") [^](#comment-1391662 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) My bad, thanks!! [→](#) [Reply](#) |\n| ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n| » [Krish-006-19](https://codeforces.com/profile/Krish-006-19 \"Newbie Krish-006-19\") | 4 days ago, [show (+1)](#) # |\n| ----------------------------------------------------------------------------------- | ---------------------------- |\n\n| » [ ![](https://codeforces.com/userpic.codeforces.org/5268131/avatar/fa8bb1ed65c58067.jpg) ](https://codeforces.com/profile/Krish-006-19) [Krish-006-19](https://codeforces.com/profile/Krish-006-19 \"Newbie Krish-006-19\") | 4 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156529?#comment-1391198 \"Link to comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) B felt like a bullet stuck in the ribs, nor does in come out nor was it blocked, felt suffocated! [→](#) [Reply](#) |\n| --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n* | » » [dolaisubhrabijoy](https://codeforces.com/profile/dolaisubhrabijoy \"Newbie dolaisubhrabijoy\") | 4 days ago, [show](#) # |\n| ------------------------------------------------------------------------------------------------- | ----------------------- |\n\n| » » [ ![](https://codeforces.com/userpic.codeforces.org/5059690/avatar/a8276810d3136972.jpg) ](https://codeforces.com/profile/dolaisubhrabijoy) [dolaisubhrabijoy](https://codeforces.com/profile/dolaisubhrabijoy \"Newbie dolaisubhrabijoy\") | 4 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156529?#comment-1391200 \"Link to comment\") [^](#comment-1391198 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) Tomorrow is my exam—worst decision ever to register! [→](#) [Reply](#) |\n| --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n* | » » [Malay\\_07](https://codeforces.com/profile/Malay%5F07 \"Newbie Malay_07\") | 4 days ago, [show](#) # |\n| ---------------------------------------------------------------------------- | ----------------------- |\n\n| » » [ ![](https://codeforces.com/userpic.codeforces.org/5652393/avatar/5e5e416d875077c9.jpg) ](https://codeforces.com/profile/Malay%5F07) [Malay\\_07](https://codeforces.com/profile/Malay%5F07 \"Newbie Malay_07\") | 4 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156529?#comment-1391213 \"Link to comment\") [^](#comment-1391198 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) same here bro...it sucks... [→](#) [Reply](#) |\n| ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n| » [misytnikita](https://codeforces.com/profile/misytnikita \"Newbie misytnikita\") | 4 days ago, [show (+1)](#) # |\n| -------------------------------------------------------------------------------- | ---------------------------- |\n\n| » [ ![](https://codeforces.com/userpic.codeforces.org/no-avatar.jpg) ](https://codeforces.com/profile/misytnikita) [misytnikita](https://codeforces.com/profile/misytnikita \"Newbie misytnikita\") | 4 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156529?#comment-1391211 \"Link to comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) A — easy, B, C literally 1500 [→](#) [Reply](#) |\n| ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n* | » » [ntkoo](https://codeforces.com/profile/ntkoo \"Newbie ntkoo\") | 3 days ago, [show](#) # |\n| ---------------------------------------------------------------- | ----------------------- |\n\n| » » [ ![](https://codeforces.com/userpic.codeforces.org/no-avatar.jpg) ](https://codeforces.com/profile/ntkoo) [ntkoo](https://codeforces.com/profile/ntkoo \"Newbie ntkoo\") | 3 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156529?#comment-1391499 \"Link to comment\") [^](#comment-1391211 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) l agree [→](#) [Reply](#) |\n| --------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ |\n\n| » [ggq15plus2](https://codeforces.com/profile/ggq15plus2 \"Candidate Master ggq15plus2\") | 4 days ago, [show (+1)](#) # |\n| --------------------------------------------------------------------------------------- | ---------------------------- |\n\n| » [ ![](https://codeforces.com/userpic.codeforces.org/no-avatar.jpg) ](https://codeforces.com/profile/ggq15plus2) [ggq15plus2](https://codeforces.com/profile/ggq15plus2 \"Candidate Master ggq15plus2\") | 4 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156529?#comment-1391216 \"Link to comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) I guess I shall improve my implementation skills now, as I got WA on #2 5 times during contest while getting AC only require changing a few chars (my specific implementation was inferior to the official solution that made debugging harder, though). Submission: [389954784](https://codeforces.com/contest/2260/submission/389954784). [→](#) [Reply](#) |\n| ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n* | » » [ggq15plus2](https://codeforces.com/profile/ggq15plus2 \"Candidate Master ggq15plus2\") | 4 days ago, [show](#) # |\n| ----------------------------------------------------------------------------------------- | ----------------------- |\n\n| » » [ ![](https://codeforces.com/userpic.codeforces.org/no-avatar.jpg) ](https://codeforces.com/profile/ggq15plus2) [ggq15plus2](https://codeforces.com/profile/ggq15plus2 \"Candidate Master ggq15plus2\") | 4 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156529?#comment-1391220 \"Link to comment\") [^](#comment-1391216 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) +11 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) I wrongly enumerated bitmasks starting from 0, yet another trivial bug in determining connectivity prevented me from detecting it from the samples :( [→](#) [Reply](#) |\n| --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ |\n* | » » [lightningmcqueen1](https://codeforces.com/profile/lightningmcqueen1 \"Newbie lightningmcqueen1\") | 4 days ago, [show (+1)](#) # |\n| ---------------------------------------------------------------------------------------------------- | ---------------------------- |\n\n| » » [ ![](https://codeforces.com/userpic.codeforces.org/3662492/avatar/c4abb8459ba1b874.jpg) ](https://codeforces.com/profile/lightningmcqueen1) [lightningmcqueen1](https://codeforces.com/profile/lightningmcqueen1 \"Newbie lightningmcqueen1\") | 4 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156529?#comment-1391228 \"Link to comment\") [^](#comment-1391216 \"Parent comment\") \\| [←](#)Rev. 3 [→](#) [![Vote: I like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) In my opinion you needed to spend more time with pencil and paper. The solution only requires 50 or so lines of code.(Assuming this was for problem C) [→](#) [Reply](#) |\n| ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n  * | » » » [ggq15plus2](https://codeforces.com/profile/ggq15plus2 \"Candidate Master ggq15plus2\") | 4 days ago, [show (+1)](#) # |\n| ------------------------------------------------------------------------------------------- | ---------------------------- |\n\n| » » » [ ![](https://codeforces.com/userpic.codeforces.org/no-avatar.jpg) ](https://codeforces.com/profile/ggq15plus2) [ggq15plus2](https://codeforces.com/profile/ggq15plus2 \"Candidate Master ggq15plus2\") | 4 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156529?#comment-1391232 \"Link to comment\") [^](#comment-1391228 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) Um, but this is F. [→](#) [Reply](#) |\n| ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n    * | » » » » [lightningmcqueen1](https://codeforces.com/profile/lightningmcqueen1 \"Newbie lightningmcqueen1\") | 4 days ago, [show](#) # |\n| -------------------------------------------------------------------------------------------------------- | ----------------------- |\n\n| » » » » [ ![](https://codeforces.com/userpic.codeforces.org/3662492/avatar/c4abb8459ba1b874.jpg) ](https://codeforces.com/profile/lightningmcqueen1) [lightningmcqueen1](https://codeforces.com/profile/lightningmcqueen1 \"Newbie lightningmcqueen1\") | 4 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156529?#comment-1391239 \"Link to comment\") [^](#comment-1391232 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) Oh nvm. You did great by the way. The tutorial for F looks like a trip to hell. [→](#) [Reply](#) |\n| ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ |\n\n| » [MastPlanHai](https://codeforces.com/profile/MastPlanHai \"Candidate Master MastPlanHai\") | 4 days ago, [show](#) # |\n| ------------------------------------------------------------------------------------------ | ----------------------- |\n\n| » [ ![](https://codeforces.com/userpic.codeforces.org/4275317/avatar/5af2ed576289c3e5.jpg) ](https://codeforces.com/profile/MastPlanHai) [MastPlanHai](https://codeforces.com/profile/MastPlanHai \"Candidate Master MastPlanHai\") | 4 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156529?#comment-1391223 \"Link to comment\") \\| [←](#)Rev. 3 [→](#) [![Vote: I like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) +4 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) Solution to E without using binary search [389952685](https://codeforces.com/contest/2260/submission/389952685 \"Submission 389952685 by MastPlanHai\") [→](#) [Reply](#) |\n| --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n| » [invoker.\\_](https://codeforces.com/profile/invoker.%5F \"Pupil invoker._\") | 4 days ago, [show (+1)](#) # |\n\n...60740 bytes truncated...\n\nThe tool call succeeded but the output was truncated. Full output saved to: /Users/zayd/.local/share/opencode/tool-output/tool_098349475001yD5suS37Bk41zT\nUse the Task tool to have explore agent process this file with Grep and Read (with offset/limit). Do NOT read the full file yourself - delegate to save context.",
            "metadata": {
              "truncated": true,
              "outputPath": "/Users/zayd/.local/share/opencode/tool-output/tool_098349475001yD5suS37Bk41zT"
            },
            "title": "https://codeforces.com/blog/entry/156529 (text/markdown; charset=utf-8)",
            "time": {
              "start": 1789259976316,
              "end": 1789259977851
            }
          },
          "metadata": {
            "openai": {
              "itemId": "fc_0993e77ff62e094b016aa5f0c77ae887d180fd818174233a34"
            }
          },
          "id": "prt_098348b63001RFLG79XIoCHRxc",
          "sessionID": "ses_f67cb8336ffe2mC4uaQqoou2Am",
          "messageID": "msg_098347d63001Qcxi9vSfoVSCeq"
        },
        {
          "reason": "tool-calls",
          "type": "step-finish",
          "tokens": {
            "total": 9134,
            "input": 9102,
            "output": 32,
            "reasoning": 0,
            "cache": {
              "write": 0,
              "read": 0
            }
          },
          "cost": 0,
          "id": "prt_098349480001EPe9zUdyL3bdn6",
          "sessionID": "ses_f67cb8336ffe2mC4uaQqoou2Am",
          "messageID": "msg_098347d63001Qcxi9vSfoVSCeq"
        }
      ]
    },
    {
      "info": {
        "parentID": "msg_098347cdf001kcKM6wljElijz5",
        "role": "assistant",
        "mode": "nudge-generation",
        "agent": "nudge-generation",
        "variant": "low",
        "path": {
          "cwd": "/private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-jPCAQg",
          "root": "/"
        },
        "cost": 0,
        "tokens": {
          "total": 21189,
          "input": 12307,
          "output": 40,
          "reasoning": 10,
          "cache": {
            "write": 0,
            "read": 8832
          }
        },
        "modelID": "gpt-6-astra",
        "providerID": "openai",
        "time": {
          "created": 1789259977863,
          "completed": 1789259983316
        },
        "finish": "tool-calls",
        "id": "msg_098349487001295QfddZuRfsfA",
        "sessionID": "ses_f67cb8336ffe2mC4uaQqoou2Am"
      },
      "parts": [
        {
          "type": "step-start",
          "id": "prt_098349cf2001Hl3vTM16Fw3EnU",
          "sessionID": "ses_f67cb8336ffe2mC4uaQqoou2Am",
          "messageID": "msg_098349487001295QfddZuRfsfA"
        },
        {
          "type": "reasoning",
          "text": "",
          "time": {
            "start": 1789259980025,
            "end": 1789259980776
          },
          "metadata": {
            "openai": {
              "itemId": "rs_0e6ddaefa942d452016aa5f0cbead487d19a00e427810f7be4",
              "reasoningEncryptedContent": "gAAAAABqpfDMqt18edCYEQazx3Rb5fyVaBu3YsH3HDxrGeCGliCikre35XrrNVtdOoQZaQJnS-WQhXjtCZkfzwPOjr9OjKtywlMVCo-qL8gIXXY1cteJIJRqgL9WNYhRfIoIkw-ZAy6gzO0DuK0Z1DtiT2Ihb0McD28MMdh9QqYFtulsRJj5rBbK9GFRV6V0I33Dtrfm5-Hjn3RdYp4Y9lLUrjG9g87cogBAqrxKxOcc1Dp82u-bHx-jptCSE_ZAcnuZIezCKKuYmaMlyu4MKVg_MxwDyOBgvZhl9mu__ZruxsjaxCT2ggTGf145cvrefMBPklDYSEvIzpyssqzW7X3LnavUERvQBQz_b84YYdMiLaZrTslS-M53EzSiAJNWJ9IaoYY9APoHCdRYj8BlZmVSG2Bcl2P9V_CS4o29LGi-Dt9SkH07UbxL2T0r_BcDIeaJ2_hNrBz5qoLIiI7bkSc3F9OTJWdR1uY09TrSR0qhuUr8U9rzQhrH-OkMfDzT1KDegIawE0g8FZt9Zg2B1wwm4tO0OHukaeZPioGQewUVNbdefluL-tYA785C-bkPei0Lu0_8l-VEIKYaIZDfTGyuzL62pDRPyf6op_ywh7lDbhRZ_1k9W-tDeSNkgW2BPyJRB9G4Ncv-gudwnqK5VBiX5l54q3WPCoLFC6Z0Jw7IBHleY73ZE0Nw6lwxpLT3nymk7XWLPcCPbAT3nwKlOj-HerDqrKaZChPOxf0Uw1bBiN7TNyUwFZ0qj-qyEeDhy6u0zcDltHvx2pZa2QPbYdT0y47-WxDvQSkon7DD5ITutYxSOaIPtmtL8CoRm-4E6DGJrCENY2SpM3MaQ0gyrbn43WIVZcgVsukyQiHESMPYvI-FBOim51LmOnoz_cQDdsJ3YjuwEN_KSSAeqQxkTB7El8o9S4J_N05aQ8trQEBYwFOlnB2I0KebbQe-wqDu6_EEG26tODzgGW9O26a11muX8ojPf6FLC5SOZRIA99wqidpXZj4lbIvKPUIZaGC6lLqKbcWat1KsEF8hte30tzRY_FCUiBkKF4JzasFHuBE9FwDFueVJHnZzm7XXpZGovxALy_9ZT087oRgso49n9PhO4Y1_eNXrI_7k02qZAFeeiTfSB1DejkKQuwgvEeNEkFDlp5dvXB08hw7oT19foAtNxdS1mYJvhSlBGOtuyQDeA6R3QcbpdHO96IqBDLP22JdfaPbFp-U1q5CEv88hkB9_3UcLtMtJBqS4FHW0djtF5rSBgYKxEZXYFPni28rr5hy8tBXFcwZOTPEb_Wr_u-j1r8dpYc9B8oObqFl-AF8LFpPChtIwoELC8cEz9qM16bo3KMiNC7i3UGo1JR95UN9dohoJ8mzp3A=="
            }
          },
          "id": "prt_098349cf9001pKdMIgDQU7aVCr",
          "sessionID": "ses_f67cb8336ffe2mC4uaQqoou2Am",
          "messageID": "msg_098349487001295QfddZuRfsfA"
        },
        {
          "type": "tool",
          "tool": "websearch",
          "callID": "call_k4tlfdVRXxllnnNVXDbGpasi",
          "state": {
            "status": "completed",
            "input": {
              "query": "2260F Edge Three Coloring cycle xor solution",
              "numResults": 5,
              "contextMaxCharacters": 4000
            },
            "output": "Title: Four Colour Theorem - A Computational View\nURL: https://www.cse.iitb.ac.in/~aad/Four-Colour.pdf\nPublished: N/A\nAuthor: Ajit Diwan\nHighlights:\nSuppose a plane triangulation is four colourable. \nAssume the four colours are 0, 1, 2, 3. \nAssign to each edge the XOR of the colours assigned to its \nendpoints. \nFour colouring implies every edge gets one of the three \ncolours 1, 2, 3. \nThe XOR of colours of edges in any cycle, and more \ngenerally, any subgraph with all vertices of even degree, \nmust be 0. \nAll three edges on the boundary of any face get distinct \ncolours.\n...\n## Three Colouring Edges\n...\nConverse holds for plane triangulations. \nConsider a colouring of edges with colours 1,2,3 such that \nthe boundary of every face has distinct colours. \nThe XOR of the colours on the boundary of any face is 0. \nAny cycle, considered as a set of edges, is the XOR of the \nboundaries of faces contained inside it. \nFor any cycle, and hence for any even subgraph, the XOR \nof the edge colours is 0. \nFor any two vertices, all paths between them must have \nthe same XOR of edge colours. \nAssign colour 0 to a fixed vertex r, and colour any other \nvertex by the XOR of the edge colours in any path from r. \nThis gives a four colouring. \n\n## Three Edge Colouring\n...\ncolouring with three colours such that the boundary of any face has distinct colours. Assign value\n...\n1 to an internal face if the colours 1,2,3 appear on the face in clockwise order and − 1 otherwise. For every internal vertex v the sum of values of faces whose boundary contains v is divisible by 3. The converse is also true, given such an assignment of +1 or − 1 values to the faces, a three edge coloring with distinct colours on the boundary of each face can be constructed.\n...\ntriangle. \nTheorem (Whitney) \nEvery plane triangulation\n...\na separating triangle has a\n...\nof the triangulation are divided into\n...\nFixing the colours\n...\nif a three\n...\nis possible. \nThe\n...\nreduces to finding a colouring of the edges\n...\nthe cycle, so\n\n---\n\nTitle: Editorial for DMOPC '19 Contest 5 P5 - Crazy Cyclic Coincidences - DMOJ: Modern Online Judge\nURL: https://dmoj.ca/problem/dmopc19c5p5/editorial\nPublished: N/A\nAuthor: N/A\nHighlights:\nNotice for this subtask that for a path to start and end at the same city, the xor-sum of the edges used must be equal to either 0 or the xor-sum of the cycle. Check if V is equal to either of them.\n...\nFor the second subtask, we observe that there always exists a subset of cycles that we shall refer to as a basis such that any path that starts and ends at the same city is equivalent to the xor-sum of exactly one subset of the basis cycles. One such basis can be created by first forming a rooted spanning tree, and for each edge not part of the spanning tree, we add to the basis the cycle that consists of the xor of the edge and the tree paths from the root to the two endpoints of the edge. Afterwards, the value of each of the basis cycles can be determined. Finally, we can check whether or not the value V, can be obtained from the xor-sum of a subset of these basis cycles by dynamic programming.\n...\nFor the third subtask, we observe that while doing the dynamic programming in Subtask 2, that a cycle can only contribute to the set of achievable xor sums if its value cannot be obtained by the xor-sum of a subset of the already considered cycles. This can be checked in O(1) time while doing the dynamic programming. Furthermore, each time such a cycle is added, it can be proven that the total number of reachable values by xor-sum must be doubled. Thus, at most O(log⁡V) of the basis cycles are going to matter. These observations are sufficient to solve this problem.\n...\nTime complexity: O(M+Vlog⁡V) or O(M+V)\n\n---\n\nTitle: Day 4 : 2000 rated problem - C. By the Assignment : r/codeforces\nURL: https://www.reddit.com/r/codeforces/comments/1qk5ujr/day_4_2000_rated_problem_c_by_the_assignment/\nPublished: 2026-01-22T20:30:31.000Z\nAuthor: u/just__observe\nHighlights:\nThe question was to simply find in how many ways we can fill up the `-1` weights of the vertices so that the graph is **balanced**. The condition for being balanced is that for any 2 vertices, any path we take between them should have the same XOR sum of weights (including the start and end vertices).\n...\n**Elimination Logic:** Now, let's say we take a cycle. Take any two adjacent nodes `u` and `v`.\n...\n1. Path 1 is just the direct edge: Value is `u ^ v` (XOR).\n2. Path 2 is the long way around the cycle.\n...\nFor the condition to hold, the XOR of the rest of the cycle must cancel out. If you do the math, this forces every vertex in that cycle (and by extension, the 2-edge-connected component) to have the **same value**.\n...\n* **Odd Parity (Odd Cycle):** If the component has an odd cycle, the math forces the value to be **0**. (Because `x ^ x = 0`, but `x ^ x ^ x = x`, so for them to match, `x` must be 0).\n* **Even Parity (Bipartite):** If it has no odd cycles, then **any value** from\n\n---\n\nTitle: Edge coloring a cycle plus triangles graph and a stronger ...\nURL: https://mathoverflow.net/questions/331566/edge-coloring-a-cycle-plus-triangles-graph-and-a-stronger-problem\nPublished: 2019-05-15T00:00:00.000Z\nAuthor: EGME\nHighlights:\nLet $G$ be a simple “cycle plus triangles” graph, that is, a graph with $3k$ vertices, $k>1$, the edges of which can be partitioned into a set that induces a $3k$\\-circuit, together with sets that induce disjoint triangles (3-circuits). Note $G$ is 4-regular.\n...\n> **QUESTION:** Is $G$ class 1, that is, can it be edge-4-colored, if it has an even number of triangles?\n> \n> **ANSWER: (2019 november 23)** **NOT** in general ... see answer 2. It remains to find an example which is 3 connected. Also, the complexity of finding the chromatic index of these graphs is to be determined ... it should be NP complete, perhaps???\n...\n(SIDE) FACT : If $G$ has an odd number of triangles then it is class 2. This is because a cycle plus triangles graph with an odd number of triangles is a regular graph of odd order ... all such graphs are class 2 (suppose it is class 1 ... then each vertex is incident with an edge of each color. But the edges of a given color are independent, so they are incident with an even number of vertices, while there is an odd number of vertices, contradiction).\n...\nCycle plus triangles graphs\n...\nknown when Erdös posed the “cycle plus\n...\nproblem” (whether\n...\nThis was solved\n...\nIn the generalization of this question to “cycle plus even $k$\\-cliques” (I hope it is clear what this means) the answer is that they are all class 1, as you can edge color the edges of the $k$\\-cliques with $k$\\-1 colors, and use two remaining colors for the edges of the even cycle in the ($k$+1)-regular graph.\n...\nI have a partial answer and I am exploring the problem with a computer now. In the positive direction, Fournier’s theorem provides a partial answer.\n...\nFournier’s theorem is a strengthening of Vizing’s theorem and it goes as follows: Let $D$ be the max degree of your graph $G$. If the vertices of $G$ that have degree $D$ are independent, then $G$ is class 1. Let us say that a graph which satisfies this condition is a Fournier graph.\n...\nSo consider a cycle plus triangles graph $G$ and vertex-3-color $G$ with colors $r, g, b$. Delete the triangle edges between vertices colored $r, b$. Then, the resulting graph is Fournier with max degree 4, so it can be edge-4-colored. So the question remains whether it is possible to reinsert the deleted edges so that $G$ remains class 1. It turns out, not always (not if there are an odd number of triangles, see the updated question).\n...\nAll examples with even number of triangles that I have explored are class 1. Recall that we are considering only simple graphs.\n...\nHere is an example that shows the answer to original question is no. The graph shown has chromatic index 5.\n...\nIn retrospect, this should have been found earlier. It remains to find a 3-connected counterexample and to establish the complexity of finding the chromatic index of these graphs\n\n---\n\nTitle: [2608.22870] Three-edge-coloring apex cubic graphs\nURL: https://arxiv.org/abs/2608.22870\nPublished: 2026-08-24T03:36:28.000Z\nAuthor: N/A\nHighlights:\n> Abstract:\n> A graph\n> G\n> is \\emph{apex} if\n> G\n> has a vertex\n> v\n> such that\n> G-v\n> is planar. We prove that every\n> 2\n> -connected apex cubic graph is three-edge-colorable. This result gives the final piece of the proof for the well-known Tutte's three-edge-coloring conjecture from 1966 \\cite{tutte}. The proof, as well as the result, generalizes that of the Four Color Theorem, which requires computer checks.\n...\n> As in the previous proof of the Four Color Theorem, the proof is constructive. More precisely, given a\n> 2\n> -connected apex cubic graph\n> G\n> on\n> n\n> vertices, our reducibility and discharging procedure yields a three-edge-coloring of\n> G\n> in\n> O(n^2)\n> time.\n...\n|Comments:|We provide detailed pseudocode specifying the computer-assisted parts of the proof. Each pseudocode is linked to the corresponding function in the source code in GitHub [this https URL](https://github.com/three-edge-coloring-apex-cubic-graphs)|",
            "metadata": {
              "provider": "exa",
              "truncated": false
            },
            "title": "Exa Web Search: 2260F Edge Three Coloring cycle xor solution",
            "time": {
              "start": 1789259981632,
              "end": 1789259983302
            }
          },
          "metadata": {
            "openai": {
              "itemId": "fc_0e6ddaefa942d452016aa5f0ccaed887d18ef0226445379282"
            }
          },
          "id": "prt_098349ff2001PPmrkkunsdNCuC",
          "sessionID": "ses_f67cb8336ffe2mC4uaQqoou2Am",
          "messageID": "msg_098349487001295QfddZuRfsfA"
        },
        {
          "reason": "tool-calls",
          "type": "step-finish",
          "tokens": {
            "total": 21189,
            "input": 12307,
            "output": 40,
            "reasoning": 10,
            "cache": {
              "write": 0,
              "read": 8832
            }
          },
          "cost": 0,
          "id": "prt_09834a9cd001qy3Gr98dOmZYzk",
          "sessionID": "ses_f67cb8336ffe2mC4uaQqoou2Am",
          "messageID": "msg_098349487001295QfddZuRfsfA"
        }
      ]
    },
    {
      "info": {
        "parentID": "msg_098347cdf001kcKM6wljElijz5",
        "role": "assistant",
        "mode": "nudge-generation",
        "agent": "nudge-generation",
        "variant": "low",
        "path": {
          "cwd": "/private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-jPCAQg",
          "root": "/"
        },
        "cost": 0,
        "tokens": {
          "total": 24139,
          "input": 2698,
          "output": 488,
          "reasoning": 89,
          "cache": {
            "write": 0,
            "read": 20864
          }
        },
        "modelID": "gpt-6-astra",
        "providerID": "openai",
        "time": {
          "created": 1789259983321,
          "completed": 1789260005016
        },
        "finish": "tool-calls",
        "id": "msg_09834a9d9001EwfynuF3g4YYPU",
        "sessionID": "ses_f67cb8336ffe2mC4uaQqoou2Am"
      },
      "parts": [
        {
          "type": "step-start",
          "id": "prt_09834ba75001XmKQ6f9rc2Q9rH",
          "sessionID": "ses_f67cb8336ffe2mC4uaQqoou2Am",
          "messageID": "msg_09834a9d9001EwfynuF3g4YYPU"
        },
        {
          "type": "reasoning",
          "text": "**Preparing exhaustive validation**",
          "time": {
            "start": 1789259987576,
            "end": 1789259993842
          },
          "metadata": {
            "openai": {
              "itemId": "rs_029234a40fbbe82e016aa5f0d3895487d2b98643817b0ce37d",
              "reasoningEncryptedContent": "gAAAAABqpfDZMzcEhzXp0b32tNNMoBkNkHDJ8JXguq4gB9ZtvP7iIHvlgxGwnhkj14DJ_TNoA2XqFmk5p_iNmjIuyKnt6_yPVwvFh_mhj39FR_MPET5RFgy3uz52UaE31l8uQftVR7EuhdddGhlqBlfnSvYiYaFnrSVDh9mD-eEEH3hF1VfXjUJbqYdf9l_dZlrIKWyjiUYzOF2mJU2Rz9rUFmIwgKFUQSCSVHixP265fGjcI02zeCx2mYDEKs49ckEP0HkRcUFxsRuyrm8ERvttNfV24W1LkwW4Fb7KMexXG4v_u8PyPiUYdCUnCqzUmX-g3r1e4OP7jjDuq5o2TfVSHLOygxgSUhMbjLfMXT1pMQNuDsHXpq6_1wKDTvMI54XAEJdNmiadN8lOmstLu24Rq4Xh3D9W_042yKfwkW7_9H9kZoHAIwn0kbndtu6g6YpYK0LfskVGXMmYkY1N-MNu_5WUu3nIQXuRVUEEocizvyNcTg4lBHunTlDhKwRNyby9ZwZzBfslhedo5nkxQmuyc0LDGFQzffwdqWueDD_pT3kzsRfx59WOVi3XG0hjmwcLcykrVEPiG1SsnH_ykDcdO7Dcbab49rtfRy63hPd9f1oJe6OxuXoM9y_ZxExpo74549k44G0vpMyxbnWifdP4UCYZJL631lm8XSbyYa1omG3ozu88r-IC13B1a9gjyXbvH4AXieqJqaePnqom8FJiCBcCDZHTzrt8O3C_Dy8VkJpA_cQyzTcIvrxYpcJs78opFECrB-QyMfUsE_u3NMUHZ_WRjvgclOvSwnsAZUdrzl0fsPxA5mJ-WDaG1r6JuDpTtn-7kiYrZ6gob73D4dzEsOHju1kFyRhwkTGlZdZtTMROiy5Qp8kSGYpvQw_k458WvfFq8oCcNPJYXQVYKt5G4gGNDNkYdjRGQqRI35qSOq74a3S85DAqIb87uFaZ1RPfkbdedtS6mqQAP5CCsKdsFEj4epA8QBAnv0iXD2061eQpnCyOAiVlnhtxBRIXUFeosB-t1oNMLLizsmr1UHZsukAvQb10xF4fBfps8VDQKCFhZTWmFDQtwhMMPNItevfNYTimbdrscC51TsuaPxeU0pAnAu-5Au992w0bhHDBPBtZWCCkvYTHAagXRy5ZsIpyS29JEeoYbXch1wCoEwek3qYM8WrjO1rgfi8EUP640NcdTx7vus5LRrZ3gfFXR7jzMRotjAXXZ3A2KvvmZRDWiTq1AElUDGLaC0asXL8Y2ctQzXqSUT5WPyXbyM_loEs5_ktXW-P83RnFQ3U6Z9u_aiFN5TI1OokoxpnBnrkltuRhUj6m2WrdWl55Fwis2npq10qr3bLGLh5-ZGR2p0PpkXPJQXNHBqetFV6z3qg6UXCRHwc1qPN3jTCK-9_x4EPbxcCvOcrCq4WpUz8DiK_RLsUUD1vPlCUuYjqr4VHNJzq3H_Cm0sVuoQkDYzSwL9BEKCO448r7yFRdKwpSAt60eBKmMXV6BhGRbvjurPeZaibRvUfW2uFk99zCzONJNGYJQk6B8mdd2tvFmy49-H8Q93nDU1ZxhE3kqsdjeTUClTWph2gvGjyH8T6gqJY55ib1wYRNO297x8Bs1ISvvTB3aKdiTkzT4PHHD--WlZYeDLskaA2OJzP8BhwUwe9DcRdY-OIvqjPJCd0kshTQYj7KaOjasPdD8NSpfT7ja1X18FeNQgw3IJjHq_j01TSKOVPqD00bTYNMJCOUAUEVq8D5OZYiGFZ4aS3bVKT_Uj6o-giAtEOrlTQMY-juPiPXonOqB4-14u0Muu6pqhW1DKAALoSaTZj5ZA=="
            }
          },
          "id": "prt_09834ba78001U0WhE2HctLnzVg",
          "sessionID": "ses_f67cb8336ffe2mC4uaQqoou2Am",
          "messageID": "msg_09834a9d9001EwfynuF3g4YYPU"
        },
        {
          "type": "tool",
          "tool": "bash",
          "callID": "call_N0LpczyWse7Rmg37Ta5o4T74",
          "state": {
            "status": "completed",
            "input": {
              "command": "python3 - <<'PY'\nfrom itertools import combinations, product\n\ndef connected(n, es):\n    seen={0}\n    while True:\n        old=len(seen)\n        for u,v in es:\n            if u in seen or v in seen: seen.update((u,v))\n        if len(seen)==old: return len(seen)==n\n\ndef brute(n, es):\n    for colors in product(range(3), repeat=len(es)):\n        if len(set(colors))<3: continue\n        ps=[list(range(n)) for _ in range(3)]\n        def root(p,x):\n            while p[x]!=x: x=p[x]\n            return x\n        for (u,v),c in zip(es,colors):\n            p=ps[c]; p[root(p,u)]=root(p,v)\n        if all(not(root(ps[c],u)==root(ps[c],v)) or root(ps[c+1],u)==root(ps[c+1],v) for c in range(2) for u in range(n) for v in range(n)):\n            return True\n    return False\n\ndef candidate(n, es):\n    for mask in range(1,1<<len(es)):\n        deg=[0]*n\n        for i,(u,v) in enumerate(es):\n            if mask>>i&1: deg[u]^=1; deg[v]^=1\n        if not any(deg) and connected(n,[e for i,e in enumerate(es) if not(mask>>i&1)]): return True\n    return False\ncount=0\nfor n in range(2,5):\n    all_es=list(combinations(range(n),2))\n    for mask in range(1<<len(all_es)):\n        es=[e for i,e in enumerate(all_es) if mask>>i&1]\n        if connected(n,es):\n            assert brute(n,es)==candidate(n,es), (n,es)\n            count+=1\nprint('Exhaustive coloring comparison passed for',count,'connected labeled graphs with n <= 4.')\nPY"
            },
            "output": "Exhaustive coloring comparison passed for 43 connected labeled graphs with n <= 4.\n",
            "metadata": {
              "output": "Exhaustive coloring comparison passed for 43 connected labeled graphs with n <= 4.\n",
              "exit": 0,
              "truncated": false
            },
            "title": "python3 - <<'PY'\nfrom itertools import combinations, product\n\ndef connected(n, es):\n    seen={0}\n    while True:\n        old=len(seen)\n        for u,v in es:\n            if u in seen or v in seen: seen.update((u,v))\n        if len(seen)==old: return len(seen)==n\n\ndef brute(n, es):\n    for colors in product(range(3), repeat=len(es)):\n        if len(set(colors))<3: continue\n        ps=[list(range(n)) for _ in range(3)]\n        def root(p,x):\n            while p[x]!=x: x=p[x]\n            return x\n        for (u,v),c in zip(es,colors):\n            p=ps[c]; p[root(p,u)]=root(p,v)\n        if all(not(root(ps[c],u)==root(ps[c],v)) or root(ps[c+1],u)==root(ps[c+1],v) for c in range(2) for u in range(n) for v in range(n)):\n            return True\n    return False\n\ndef candidate(n, es):\n    for mask in range(1,1<<len(es)):\n        deg=[0]*n\n        for i,(u,v) in enumerate(es):\n            if mask>>i&1: deg[u]^=1; deg[v]^=1\n        if not any(deg) and connected(n,[e for i,e in enumerate(es) if not(mask>>i&1)]): return True\n    return False\ncount=0\nfor n in range(2,5):\n    all_es=list(combinations(range(n),2))\n    for mask in range(1<<len(all_es)):\n        es=[e for i,e in enumerate(all_es) if mask>>i&1]\n        if connected(n,es):\n            assert brute(n,es)==candidate(n,es), (n,es)\n            count+=1\nprint('Exhaustive coloring comparison passed for',count,'connected labeled graphs with n <= 4.')\nPY",
            "time": {
              "start": 1789260004928,
              "end": 1789260005012
            }
          },
          "metadata": {
            "openai": {
              "itemId": "fc_029234a40fbbe82e016aa5f0d99f8087d28ac65dea84ea7a77"
            }
          },
          "id": "prt_09834d2f7001HruofMuLUAmB67",
          "sessionID": "ses_f67cb8336ffe2mC4uaQqoou2Am",
          "messageID": "msg_09834a9d9001EwfynuF3g4YYPU"
        },
        {
          "reason": "tool-calls",
          "type": "step-finish",
          "tokens": {
            "total": 24139,
            "input": 2698,
            "output": 488,
            "reasoning": 89,
            "cache": {
              "write": 0,
              "read": 20864
            }
          },
          "cost": 0,
          "id": "prt_09834fe960013Qh0ur4TrXaSuB",
          "sessionID": "ses_f67cb8336ffe2mC4uaQqoou2Am",
          "messageID": "msg_09834a9d9001EwfynuF3g4YYPU"
        }
      ]
    },
    {
      "info": {
        "parentID": "msg_098347cdf001kcKM6wljElijz5",
        "role": "assistant",
        "mode": "nudge-generation",
        "agent": "nudge-generation",
        "variant": "low",
        "path": {
          "cwd": "/private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-jPCAQg",
          "root": "/"
        },
        "cost": 0,
        "tokens": {
          "total": 26867,
          "input": 872,
          "output": 2607,
          "reasoning": 92,
          "cache": {
            "write": 0,
            "read": 23296
          }
        },
        "modelID": "gpt-6-astra",
        "providerID": "openai",
        "time": {
          "created": 1789260005018,
          "completed": 1789260090863
        },
        "finish": "tool-calls",
        "structured": {
          "status": "success",
          "reason": null,
          "hints": [
            {
              "order": 1,
              "content": "Look at the connected components formed by color $3$. Can any edge of color $1$ or $2$ connect two different such components? What does the connectivity of the original graph then force?"
            },
            {
              "order": 2,
              "content": "Pick a color-$1$ edge $(u,v)$. There must be a color-$2$ path from $u$ to $v$, so together they contain a cycle. Deleting that cycle leaves all color-$3$ edges intact. Conversely, if deleting a cycle keeps the graph connected, try coloring one cycle edge $1$, the rest of the cycle $2$, and everything else $3$."
            },
            {
              "order": 3,
              "content": "You need a cycle whose edges can all be deleted without disconnecting the graph. It is safe to search a slightly larger family: nonempty edge sets in which every vertex has even degree. Such a set contains a cycle, and deleting just that cycle removes fewer edges."
            },
            {
              "order": 4,
              "content": "Choose a spanning tree. There are only $k=m-n+1\\le 10$ non-tree edges. Each non-tree edge, together with the tree path between its endpoints, forms a fundamental cycle. Every even-degree edge set is an XOR of some of these $k$ cycles, so only $2^k-1$ nonempty candidates are needed."
            },
            {
              "order": 5,
              "content": "For each edge $e$, store a $k$-bit mask $b_e$ indicating which fundamental cycles contain it. Under a chosen subset $s$ of cycles, delete $e$ exactly when $\\operatorname{popcount}(b_e\\mathbin{\\&}s)$ is odd. Run BFS or DFS using the remaining edges. If any nonzero $s$ leaves all vertices connected, answer `YES`; otherwise answer `NO`. This takes $O(k n+2^k(n+m))$ time."
            }
          ],
          "editorial": "The title is a bit of a trap: adjacent edges are allowed to have the same color. This is a connectivity problem, not standard edge coloring.\n\nThe useful small parameter is\n$$k=m-(n-1)\\le 10,$$\nthe number of edges outside a spanning tree.\n\n## 1. Turn the coloring into a cycle\n\n**In every valid coloring, the color-$3$ edges connect the whole graph.**\n\nFor any original edge $(u,v)$:\n- If its color is $3$, its endpoints are directly connected in color $3$.\n- If its color is $1$ or $2$, the condition also makes its endpoints connected in color $3$.\n\nThus every original edge stays inside a single color-$3$ component. Since the original graph is connected, there can only be one such component.\n\nNow choose a color-$1$ edge $e=(u,v)$. Its endpoints must have a color-$2$ path between them. Choose a simple such path $P$. The edge $e$ and the path $P$ form a simple cycle $C$.\n\nDeleting all edges of $C$ leaves the graph connected: none of them had color $3$, so the connected color-$3$ subgraph survives.\n\nThe converse also works. Suppose a simple cycle $C$ can be deleted without disconnecting the graph. Color:\n- One edge of $C$ with color $1$.\n- Every other edge of $C$ with color $2$.\n- Every edge outside $C$ with color $3$.\n\nThe color-$2$ path connects the endpoints of the only color-$1$ edge. Color $3$ connects every pair of vertices. All three colors appear: a simple cycle has at least three edges, and its connected complement has at least $n-1\\ge 1$ edges.\n\nSo we have an exact characterization:\n\n> The answer is `YES` if and only if some simple cycle can be deleted while leaving the graph connected.\n\n## 2. Search even-degree edge sets instead\n\nEnumerating simple cycles directly is awkward. We can search a larger, much easier family.\n\nCall an edge set $S$ **even** if every vertex has even degree using only edges from $S$. Isolated vertices have degree zero, so they are fine.\n\nA simple cycle is even. Conversely, every nonempty even edge set contains a simple cycle: in any component containing an edge, all non-isolated vertices have degree at least two, which guarantees a cycle.\n\nIf deleting $S$ leaves the graph connected, choose a cycle $C\\subseteq S$. Deleting only $C$ also leaves it connected, because we remove fewer edges.\n\nTherefore, our task is equivalently:\n\n> Find a nonempty even edge set whose deletion leaves the graph connected.\n\nWe do **not** need the candidate itself to be one simple cycle.\n\n## 3. Only ten independent cycle choices\n\nBuild any spanning tree $T$. For each non-tree edge $e_i=(u_i,v_i)$, define its **fundamental cycle** $C_i$: the edge $e_i$ plus the unique tree path between its endpoints.\n\nThere are exactly $k=m-n+1\\le 10$ such cycles.\n\nXOR of edge sets means keeping edges that appear an odd number of times. Every XOR of fundamental cycles is even, because degrees remain even under XOR.\n\nMore importantly, **every even edge set is such an XOR**. Here is the full argument:\n\n1. Take an even edge set $S$.\n2. For every non-tree edge $e_i$ belonging to $S$, XOR $S$ with $C_i$.\n3. This removes all non-tree edges. Each fundamental cycle contains its own non-tree edge and no other non-tree edge.\n4. The remaining set is still even and uses only tree edges.\n5. A nonempty subset of tree edges is a forest with a degree-$1$ vertex in some nontrivial component. It cannot be even. Therefore, the remaining set is empty.\n\nThis proves the representation. It also shows that any nonempty subset of fundamental cycles has nonempty XOR: a selected cycle's unique non-tree edge cannot cancel.\n\nWe can now enumerate all masks from $1$ to $2^k-1$. At most $1023$ candidates. Brute force finally has a sensible target.\n\n## 4. Represent all cycles with one mask per edge\n\nFor each edge $e$, store a bitmask $b_e$. Bit $i$ is set when $e$ belongs to $C_i$.\n\nTo construct cycle $i$:\n- Toggle bit $i$ on its non-tree edge $(u_i,v_i)$.\n- Walk from $u_i$ to the tree root, toggling bit $i$ on every parent edge.\n- Do the same from $v_i$.\n\nThe shared part of the two root paths is toggled twice and cancels, leaving exactly the tree path between the endpoints.\n\nFor a subset mask $s$, edge $e$ belongs to the XOR precisely when\n$$\\operatorname{popcount}(b_e\\mathbin{\\&}s)\\equiv 1\\pmod 2.$$\n\nRun BFS from vertex $0$, skipping exactly those edges. If BFS visits all $n$ vertices, print `YES`. If no nonzero mask succeeds, print `NO`.\n\n**Do not test mask zero.** It deletes nothing, so it would accept every connected graph without providing any cycle—or any way to use all three colors.\n\n## Correctness\n\nIf the algorithm prints `YES`, it has found a nonempty XOR of fundamental cycles whose deletion preserves connectivity. This is a nonempty even edge set, so it contains a simple cycle whose deletion also preserves connectivity. The coloring construction above gives a valid coloring using all three colors.\n\nIf a valid coloring exists, the first reduction gives a simple cycle whose deletion preserves connectivity. That cycle is even, so it is the XOR of a nonempty subset of the fundamental cycles. The algorithm enumerates this subset, finds the remaining graph connected, and prints `YES`.\n\nThus the algorithm prints `YES` exactly when a valid coloring exists.\n\n## Complexity\n\nBuilding the spanning tree takes $O(n+m)$ time. Constructing the edge masks takes $O(kn)$ time using the root walks. Each of the $2^k-1$ candidates needs one $O(n+m)$ connectivity check.\n\nTotal time is\n$$O\\bigl(kn+2^k(n+m)\\bigr),$$\nwith $O(n+m)$ space. Since $k\\le 10$ and the total number of vertices is at most $3000$, this comfortably fits the limits.",
          "solution": "#include <bits/stdc++.h>\nusing namespace std;\n\nusing ll = long long;\n\nvoid setIO() {\n    ios::sync_with_stdio(false);\n    cin.tie(nullptr);\n}\n\nvoid solve() {\n    int n, m;\n    cin >> n >> m;\n    vector<pair<int, int>> edges(m);\n    vector<vector<pair<int, int>>> g(n);\n    for (int i = 0; i < m; ++i) {\n        auto& [u, v] = edges[i];\n        cin >> u >> v;\n        --u;\n        --v;\n        g[u].push_back({v, i});\n        g[v].push_back({u, i});\n    }\n\n    vector<int> parent(n, -1), pe(n, -1), q(n);\n    vector<char> tree(m, false);\n    parent[0] = 0;\n    q[0] = 0;\n    int size = 1;\n    for (int head = 0; head < size; ++head) {\n        int u = q[head];\n        for (auto [v, id] : g[u]) {\n            if (parent[v] != -1) continue;\n            parent[v] = u;\n            pe[v] = id;\n            tree[id] = true;\n            q[size++] = v;\n        }\n    }\n\n    vector<unsigned> basis(m, 0);\n    int k = 0;\n    for (int id = 0; id < m; ++id) {\n        if (tree[id]) continue;\n        unsigned bit = 1u << k++;\n        basis[id] ^= bit;\n        auto [u, v] = edges[id];\n        for (int x = u; x != 0; x = parent[x])\n            basis[pe[x]] ^= bit;\n        for (int x = v; x != 0; x = parent[x])\n            basis[pe[x]] ^= bit;\n    }\n\n    vector<char> seen(n);\n    for (unsigned mask = 1; mask < (1u << k); ++mask) {\n        fill(seen.begin(), seen.end(), false);\n        seen[0] = true;\n        q[0] = 0;\n        size = 1;\n        for (int head = 0; head < size; ++head) {\n            int u = q[head];\n            for (auto [v, id] : g[u]) {\n                if (seen[v] || (popcount(basis[id] & mask) & 1))\n                    continue;\n                seen[v] = true;\n                q[size++] = v;\n            }\n        }\n        if (size == n) {\n            cout << \"YES\\n\";\n            return;\n        }\n    }\n    cout << \"NO\\n\";\n}\n\nint main() {\n    setIO();\n    int t;\n    cin >> t;\n    while (t--) solve();\n}\n"
        },
        "id": "msg_09834fe9a001GJ5DbCC24EYgH8",
        "sessionID": "ses_f67cb8336ffe2mC4uaQqoou2Am"
      },
      "parts": [
        {
          "type": "step-start",
          "id": "prt_0983510ee001qu4N8PViM07oFu",
          "sessionID": "ses_f67cb8336ffe2mC4uaQqoou2Am",
          "messageID": "msg_09834fe9a001GJ5DbCC24EYgH8"
        },
        {
          "type": "reasoning",
          "text": "",
          "time": {
            "start": 1789260009714,
            "end": 1789260014732
          },
          "metadata": {
            "openai": {
              "itemId": "rs_01a4b27ae6481093016aa5f0e9acbc87d2b270985eaeaec4bf",
              "reasoningEncryptedContent": "gAAAAABqpfDuLtbv09GFaTlE7HSOzuZ4yDLangova-l74grQf6ir8CHNOHwjqgMLvd9QGp6B8-fI02qmEIHFu6aBvWYwLnHm_BhKYFywCX5xso4XF9wShkXqGrpN3NaJPrwr3_gPXqnLn4mv1-H14arYcxEBwUxVSzHB9pjQC2xAuxB9N24ghpzCC5VmslPQ_6S2wHwKKu-tfE2eJZIE6TZIo1RLQUXwS7iVQntRkU-DYQ57zGtXy6xeJrkS5bZfPlqRsfcbhXlyjFzp1ZHdpvLWFy0kjmq84T37JL2WP2uDBmzA-2mUtnJT8oKOLYVXBA6MH4jD48feNeEKqdtYO30OFHq9SrtH23BeI-hs2wkf0ROeN4YingpZBEAQ3ox6aa0mnxr48PJ90S1iMmHDF5J_axR6dulOvvUHFOuRnWcBUTmUAmYeAwT-SpNVAPAzVvyPrnyoCugwb2LzmZmkmZKEp7UeeVmco0yJWGxeRfpRtdk-VCYA7jdKYoV366m4IPcUu-ecbLei_uE01mtWoQgzBgqheemas3_HeGKw4KMEnAO0Nt_cP6FlRWYxzmdoLMjjIOng1RYtie-0rec76FxWGbRVH9bzv8rfOnbI-vbsQgzMqNHobdCWeFxtf-qPwVuVkXmrXiUV7IUAtw9RZWZtcLqh5FxNUFvU8Hc6yPr1JZbr_oaeDpSHmcTmbVbce9deHXGZW42vswX09IhWu7a7I8MnGjpUzwdcqumzi2u3idbjk40AVumhIRcHuzi1BxakZ8RzfQUCjuxKLDh3hDhniKaJ0ImdWi9R7kz5VklbnT423jBqA-rgjHf4oEnZy3kx264lqYsCV7U8vsTh4uw2g8pECCE27vTSjOK4Ev7wqrKzgSndxYLRY9GTxKibxAUtbJZfnBnu2p8_m4MSZDq5zIiWuFv-IvnXdbaHxAI1jB7FWpXbaa44L_w6U_39ZlCvt5RGM6PLgR4yeDcIuGM-tU0k-_Y_ApscXJfxGwx3Hm-lzQESXidmvlnxyMx20cuviF-VsOeZ--uG7i1fDR3UbpITYtSgbQfSROW28TGcBfSvf76iz38MnnsimGTumE6u5DT-xBpvUPQB3Zv0SgkODvC5T0iPrUflSjbH9fQTtRd1LlMYn3TCWEhlZDwIeOfJ7KMf6VB6FBVi3di5hMtuMmKKye055f2LxIQO1PPnxwap5izcULVeSAhtITVgKJdOTxP5uEsv7N6b575LYeJsR9rTN4YnswwFGb-4uP5hdPd2elPSJqHkYkhtLaa0YBNB-pk8PJ2a1PANXdUcHWNJ6qG0rx-wSTC61Vl6WGEIOyu_gyVuqz_gDhCAgYVBSlq0jtLvrCttvR-ww4echq5o2Ojqx7g8lnNfM3u6RLfQMMc0_Ya9k6Ahl1s9xUc2BGGKK9wqV-vzMy9Mu533CCnGOvg8EBUZRlr0rR2cRgwL4Q-_OOLwiULxXLQHxe8wTmynM2r8oho9qufvLoNUOJCFN3HhDE_2K7FK7bYOOnBFXF9AiVCHNqo_NEsYM92qNmh002dGdnmh9UAkc_SgBVb6n3yL0ugvCsf8QSlR0JnVfKDhuGwRPdqU3yhdNwWEssZrTnAw3IBqYGuJ_pAILWoACCF3524WCiY8llLWDCGwagvSy6iLb_LokFBljpL3vPocG9Ssk2tJIgD_dUSvLLpF1IjNvkljj_neCnTs--pVNReXGGZ5H5_xi5eKpoeRNsr6KtpW6NgdaLEDr5VIDOxJrKKLgjdHncVIb1yxe54SQdTzyhBfQVghS43WjCMW8YZ-kNEivGOVUEwlGLV7ZnksvPw9bzg2Kc4crluQhd5MX2F3Dy5whQ7K-hsgJ3DwA9RbfJEbOF2c"
            }
          },
          "id": "prt_0983510f2001eBU7eDnD9al0TH",
          "sessionID": "ses_f67cb8336ffe2mC4uaQqoou2Am",
          "messageID": "msg_09834fe9a001GJ5DbCC24EYgH8"
        },
        {
          "type": "tool",
          "tool": "StructuredOutput",
          "callID": "call_nXYLfDg14qqiI3G334MUQPk6",
          "state": {
            "status": "completed",
            "input": {
              "status": "success",
              "reason": null,
              "hints": [
                {
                  "order": 1,
                  "content": "Look at the connected components formed by color $3$. Can any edge of color $1$ or $2$ connect two different such components? What does the connectivity of the original graph then force?"
                },
                {
                  "order": 2,
                  "content": "Pick a color-$1$ edge $(u,v)$. There must be a color-$2$ path from $u$ to $v$, so together they contain a cycle. Deleting that cycle leaves all color-$3$ edges intact. Conversely, if deleting a cycle keeps the graph connected, try coloring one cycle edge $1$, the rest of the cycle $2$, and everything else $3$."
                },
                {
                  "order": 3,
                  "content": "You need a cycle whose edges can all be deleted without disconnecting the graph. It is safe to search a slightly larger family: nonempty edge sets in which every vertex has even degree. Such a set contains a cycle, and deleting just that cycle removes fewer edges."
                },
                {
                  "order": 4,
                  "content": "Choose a spanning tree. There are only $k=m-n+1\\le 10$ non-tree edges. Each non-tree edge, together with the tree path between its endpoints, forms a fundamental cycle. Every even-degree edge set is an XOR of some of these $k$ cycles, so only $2^k-1$ nonempty candidates are needed."
                },
                {
                  "order": 5,
                  "content": "For each edge $e$, store a $k$-bit mask $b_e$ indicating which fundamental cycles contain it. Under a chosen subset $s$ of cycles, delete $e$ exactly when $\\operatorname{popcount}(b_e\\mathbin{\\&}s)$ is odd. Run BFS or DFS using the remaining edges. If any nonzero $s$ leaves all vertices connected, answer `YES`; otherwise answer `NO`. This takes $O(k n+2^k(n+m))$ time."
                }
              ],
              "editorial": "The title is a bit of a trap: adjacent edges are allowed to have the same color. This is a connectivity problem, not standard edge coloring.\n\nThe useful small parameter is\n$$k=m-(n-1)\\le 10,$$\nthe number of edges outside a spanning tree.\n\n## 1. Turn the coloring into a cycle\n\n**In every valid coloring, the color-$3$ edges connect the whole graph.**\n\nFor any original edge $(u,v)$:\n- If its color is $3$, its endpoints are directly connected in color $3$.\n- If its color is $1$ or $2$, the condition also makes its endpoints connected in color $3$.\n\nThus every original edge stays inside a single color-$3$ component. Since the original graph is connected, there can only be one such component.\n\nNow choose a color-$1$ edge $e=(u,v)$. Its endpoints must have a color-$2$ path between them. Choose a simple such path $P$. The edge $e$ and the path $P$ form a simple cycle $C$.\n\nDeleting all edges of $C$ leaves the graph connected: none of them had color $3$, so the connected color-$3$ subgraph survives.\n\nThe converse also works. Suppose a simple cycle $C$ can be deleted without disconnecting the graph. Color:\n- One edge of $C$ with color $1$.\n- Every other edge of $C$ with color $2$.\n- Every edge outside $C$ with color $3$.\n\nThe color-$2$ path connects the endpoints of the only color-$1$ edge. Color $3$ connects every pair of vertices. All three colors appear: a simple cycle has at least three edges, and its connected complement has at least $n-1\\ge 1$ edges.\n\nSo we have an exact characterization:\n\n> The answer is `YES` if and only if some simple cycle can be deleted while leaving the graph connected.\n\n## 2. Search even-degree edge sets instead\n\nEnumerating simple cycles directly is awkward. We can search a larger, much easier family.\n\nCall an edge set $S$ **even** if every vertex has even degree using only edges from $S$. Isolated vertices have degree zero, so they are fine.\n\nA simple cycle is even. Conversely, every nonempty even edge set contains a simple cycle: in any component containing an edge, all non-isolated vertices have degree at least two, which guarantees a cycle.\n\nIf deleting $S$ leaves the graph connected, choose a cycle $C\\subseteq S$. Deleting only $C$ also leaves it connected, because we remove fewer edges.\n\nTherefore, our task is equivalently:\n\n> Find a nonempty even edge set whose deletion leaves the graph connected.\n\nWe do **not** need the candidate itself to be one simple cycle.\n\n## 3. Only ten independent cycle choices\n\nBuild any spanning tree $T$. For each non-tree edge $e_i=(u_i,v_i)$, define its **fundamental cycle** $C_i$: the edge $e_i$ plus the unique tree path between its endpoints.\n\nThere are exactly $k=m-n+1\\le 10$ such cycles.\n\nXOR of edge sets means keeping edges that appear an odd number of times. Every XOR of fundamental cycles is even, because degrees remain even under XOR.\n\nMore importantly, **every even edge set is such an XOR**. Here is the full argument:\n\n1. Take an even edge set $S$.\n2. For every non-tree edge $e_i$ belonging to $S$, XOR $S$ with $C_i$.\n3. This removes all non-tree edges. Each fundamental cycle contains its own non-tree edge and no other non-tree edge.\n4. The remaining set is still even and uses only tree edges.\n5. A nonempty subset of tree edges is a forest with a degree-$1$ vertex in some nontrivial component. It cannot be even. Therefore, the remaining set is empty.\n\nThis proves the representation. It also shows that any nonempty subset of fundamental cycles has nonempty XOR: a selected cycle's unique non-tree edge cannot cancel.\n\nWe can now enumerate all masks from $1$ to $2^k-1$. At most $1023$ candidates. Brute force finally has a sensible target.\n\n## 4. Represent all cycles with one mask per edge\n\nFor each edge $e$, store a bitmask $b_e$. Bit $i$ is set when $e$ belongs to $C_i$.\n\nTo construct cycle $i$:\n- Toggle bit $i$ on its non-tree edge $(u_i,v_i)$.\n- Walk from $u_i$ to the tree root, toggling bit $i$ on every parent edge.\n- Do the same from $v_i$.\n\nThe shared part of the two root paths is toggled twice and cancels, leaving exactly the tree path between the endpoints.\n\nFor a subset mask $s$, edge $e$ belongs to the XOR precisely when\n$$\\operatorname{popcount}(b_e\\mathbin{\\&}s)\\equiv 1\\pmod 2.$$\n\nRun BFS from vertex $0$, skipping exactly those edges. If BFS visits all $n$ vertices, print `YES`. If no nonzero mask succeeds, print `NO`.\n\n**Do not test mask zero.** It deletes nothing, so it would accept every connected graph without providing any cycle—or any way to use all three colors.\n\n## Correctness\n\nIf the algorithm prints `YES`, it has found a nonempty XOR of fundamental cycles whose deletion preserves connectivity. This is a nonempty even edge set, so it contains a simple cycle whose deletion also preserves connectivity. The coloring construction above gives a valid coloring using all three colors.\n\nIf a valid coloring exists, the first reduction gives a simple cycle whose deletion preserves connectivity. That cycle is even, so it is the XOR of a nonempty subset of the fundamental cycles. The algorithm enumerates this subset, finds the remaining graph connected, and prints `YES`.\n\nThus the algorithm prints `YES` exactly when a valid coloring exists.\n\n## Complexity\n\nBuilding the spanning tree takes $O(n+m)$ time. Constructing the edge masks takes $O(kn)$ time using the root walks. Each of the $2^k-1$ candidates needs one $O(n+m)$ connectivity check.\n\nTotal time is\n$$O\\bigl(kn+2^k(n+m)\\bigr),$$\nwith $O(n+m)$ space. Since $k\\le 10$ and the total number of vertices is at most $3000$, this comfortably fits the limits.",
              "solution": "#include <bits/stdc++.h>\nusing namespace std;\n\nusing ll = long long;\n\nvoid setIO() {\n    ios::sync_with_stdio(false);\n    cin.tie(nullptr);\n}\n\nvoid solve() {\n    int n, m;\n    cin >> n >> m;\n    vector<pair<int, int>> edges(m);\n    vector<vector<pair<int, int>>> g(n);\n    for (int i = 0; i < m; ++i) {\n        auto& [u, v] = edges[i];\n        cin >> u >> v;\n        --u;\n        --v;\n        g[u].push_back({v, i});\n        g[v].push_back({u, i});\n    }\n\n    vector<int> parent(n, -1), pe(n, -1), q(n);\n    vector<char> tree(m, false);\n    parent[0] = 0;\n    q[0] = 0;\n    int size = 1;\n    for (int head = 0; head < size; ++head) {\n        int u = q[head];\n        for (auto [v, id] : g[u]) {\n            if (parent[v] != -1) continue;\n            parent[v] = u;\n            pe[v] = id;\n            tree[id] = true;\n            q[size++] = v;\n        }\n    }\n\n    vector<unsigned> basis(m, 0);\n    int k = 0;\n    for (int id = 0; id < m; ++id) {\n        if (tree[id]) continue;\n        unsigned bit = 1u << k++;\n        basis[id] ^= bit;\n        auto [u, v] = edges[id];\n        for (int x = u; x != 0; x = parent[x])\n            basis[pe[x]] ^= bit;\n        for (int x = v; x != 0; x = parent[x])\n            basis[pe[x]] ^= bit;\n    }\n\n    vector<char> seen(n);\n    for (unsigned mask = 1; mask < (1u << k); ++mask) {\n        fill(seen.begin(), seen.end(), false);\n        seen[0] = true;\n        q[0] = 0;\n        size = 1;\n        for (int head = 0; head < size; ++head) {\n            int u = q[head];\n            for (auto [v, id] : g[u]) {\n                if (seen[v] || (popcount(basis[id] & mask) & 1))\n                    continue;\n                seen[v] = true;\n                q[size++] = v;\n            }\n        }\n        if (size == n) {\n            cout << \"YES\\n\";\n            return;\n        }\n    }\n    cout << \"NO\\n\";\n}\n\nint main() {\n    setIO();\n    int t;\n    cin >> t;\n    while (t--) solve();\n}\n"
            },
            "output": "Structured output captured successfully.",
            "metadata": {
              "valid": true
            },
            "title": "Structured Output",
            "time": {
              "start": 1789260090735,
              "end": 1789260090740
            }
          },
          "metadata": {
            "openai": {
              "itemId": "fc_01a4b27ae6481093016aa5f0ee998887d288794924d8111352"
            }
          },
          "id": "prt_098352491001xbkW6RWbP9mbwR",
          "sessionID": "ses_f67cb8336ffe2mC4uaQqoou2Am",
          "messageID": "msg_09834fe9a001GJ5DbCC24EYgH8"
        },
        {
          "reason": "tool-calls",
          "type": "step-finish",
          "tokens": {
            "total": 26867,
            "input": 872,
            "output": 2607,
            "reasoning": 92,
            "cache": {
              "write": 0,
              "read": 23296
            }
          },
          "cost": 0,
          "id": "prt_098364deb001KXxX10ATOdUFSI",
          "sessionID": "ses_f67cb8336ffe2mC4uaQqoou2Am",
          "messageID": "msg_09834fe9a001GJ5DbCC24EYgH8"
        }
      ]
    }
  ]
}