{
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    "slug": "quick-knight",
    "projectID": "global",
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    "path": "private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-qTDVNT",
    "title": "Nudge content generation",
    "agent": "nudge-generation",
    "model": {
      "id": "gpt-6-astra",
      "providerID": "openai",
      "variant": "low"
    },
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    "summary": {
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    "tokens": {
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      "reasoning": 20,
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  "messages": [
    {
      "info": {
        "role": "user",
        "time": {
          "created": 1789698140153
        },
        "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",
                    "pattern": "^[^\\u0000]*$",
                    "minLength": 1,
                    "maxLength": 2000
                  },
                  {
                    "type": "null"
                  }
                ],
                "description": "If status is 'unsolvable', provide a detailed failure message to show users. Name the concrete statement/resource blocker and any relevant source lookup/access status, such as missing official tutorial, 403/Cloudflare challenge, 404, or unavailable accepted submissions. Otherwise null."
              },
              "hints": {
                "anyOf": [
                  {
                    "type": "array",
                    "minItems": 5,
                    "maxItems": 5,
                    "description": "Exactly 5 progressive hints, each building on the last.",
                    "items": {
                      "type": "object",
                      "properties": {
                        "order": {
                          "type": "integer",
                          "minimum": 1,
                          "maximum": 5,
                          "description": "Hint number, 1 through 5."
                        },
                        "content": {
                          "type": "string",
                          "pattern": "^[^\\u0000]*$",
                          "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",
                    "pattern": "^[^\\u0000]*$",
                    "minLength": 1,
                    "maxLength": 100000
                  },
                  {
                    "type": "null"
                  }
                ],
                "description": "If status is 'success', a prose editorial explaining the solution in Nudge's own voice, with no links, citations, or source/research notes. Otherwise null."
              },
              "solution": {
                "anyOf": [
                  {
                    "type": "string",
                    "pattern": "^[^\\u0000]*$",
                    "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_0b25267f9001dKimfBE2zpQytf",
        "sessionID": "ses_f4dad9816ffe3WT2TX5xFhSURB"
      },
      "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 2253A: \"The Best Card\" (rated 800).\n\nKnown problem metadata:\n- Contest ID: 2253\n- Index: A\n- Name: The Best Card\n- Rating: 800 (Codeforces difficulty rating)\n- Tags: greedy, math, number theory\n- Problem URL: https://codeforces.com/contest/2253/problem/A\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\">A. The Best Card</div><div class=\"time-limit\"><div class=\"property-title\">time limit per test</div>2 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>In a card game, there are $$$n$$$ cards with values $$$2, 3, 4, \\ldots, n + 1$$$.</p><p>To determine which of two cards with values $$$x$$$ and $$$y$$$ wins, apply the following rules: </p><ul> <li> if one of the numbers $$$x$$$ and $$$y$$$ is divisible by the other, the card with the smaller value wins; </li><li> otherwise, the card with the larger value wins. </li></ul><p>For example, between cards $$$2$$$ and $$$6$$$, card $$$2$$$ wins because $$$6$$$ is divisible by $$$2$$$. Between cards $$$4$$$ and $$$6$$$, card $$$6$$$ wins because neither of these numbers is divisible by the other.</p><p>Determine whether there exists a card that wins against every other card.</p></div><div class=\"input-specification\"><div class=\"section-title\">Input</div><p>The first line contains an integer $$$t$$$ ($$$1 \\le t \\le 10^4$$$)&nbsp;— the number of test cases.</p><p>The only line of each test case contains an integer $$$n$$$ ($$$2 \\le n \\le 2 \\cdot 10^5$$$)&nbsp;— the number of cards in the game.</p><p>Additional constraints on the input: </p><ul> <li> the sum of $$$n$$$ over all test cases does not exceed $$$3 \\cdot 10^6$$$. </li></ul></div><div class=\"output-specification\"><div class=\"section-title\">Output</div><p>For each test case, print <span class=\"tex-font-style-tt\">YES</span> if there is a card that wins against all other cards, and <span class=\"tex-font-style-tt\">NO</span> otherwise.</p><p>Each letter may be printed in either case. For example, <span class=\"tex-font-style-tt\">YES</span>, <span class=\"tex-font-style-tt\">yes</span>, and <span class=\"tex-font-style-tt\">yEs</span> are all recognized as a positive answer.</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\">5</div><div class=\"test-example-line test-example-line-odd test-example-line-1\">2</div><div class=\"test-example-line test-example-line-even test-example-line-2\">3</div><div class=\"test-example-line test-example-line-odd test-example-line-3\">4</div><div class=\"test-example-line test-example-line-even test-example-line-4\">5</div><div class=\"test-example-line test-example-line-odd test-example-line-5\">8</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\">YES</div><div class=\"test-example-line test-example-line-even test-example-line-4\">NO</div><div class=\"test-example-line test-example-line-odd test-example-line-5\">NO</div></pre></div></div></div><div class=\"note\"><div class=\"section-title\">Note</div><p>In the first test case, the available cards have values $$$2$$$ and $$$3$$$. Card $$$3$$$ wins against card $$$2$$$.</p><p>In the second test case, the available cards have values $$$2$$$, $$$3$$$, and $$$4$$$. Card $$$2$$$ wins against card $$$4$$$, card $$$3$$$ wins against card $$$2$$$, and card $$$4$$$ wins against card $$$3$$$, so there is no suitable card.</p></div>\n</problem-statement>\n\n\nSource lookup status:\n<source-lookup-status>\n- Tutorial: https://codeforces.com/blog/entry/155813 loaded (200 OK; title: \"Educational Codeforces Round 193 — Editorial - Codeforces\"; mentions 2253A)\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- Never include NUL (U+0000) characters in any field. Use literal $ or $$ for math delimiters, never control characters.\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 an object with `order` (1 through 5) and `content` (the Markdown hint text), not a plain string. The `content` must contain just the hint text, with 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```",
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    },
    {
      "info": {
        "parentID": "msg_0b25267f9001dKimfBE2zpQytf",
        "role": "assistant",
        "mode": "nudge-generation",
        "agent": "nudge-generation",
        "variant": "low",
        "path": {
          "cwd": "/private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-qTDVNT",
          "root": "/"
        },
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        "modelID": "gpt-6-astra",
        "providerID": "openai",
        "time": {
          "created": 1789698140292,
          "completed": 1789698144788
        },
        "finish": "tool-calls",
        "id": "msg_0b2526884001MGtZLdJOh1e6g8",
        "sessionID": "ses_f4dad9816ffe3WT2TX5xFhSURB"
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      "parts": [
        {
          "type": "step-start",
          "id": "prt_0b25272c7001J0CWHTmCnQSXCX",
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        {
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          "callID": "call_TaOwGQBHluuKmr8PpGw6TSTU",
          "state": {
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            "input": {
              "url": "https://codeforces.com/blog/entry/155813",
              "format": "markdown"
            },
            "output": "---\nimage: https://codeforces.com/codeforces.org/s/13758/images/codeforces-telegram-square-1024x1024.png\ntitle: Educational Codeforces Round 193 — 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/155813?locale=en) [![По-русски](//codeforces.com/codeforces.org/s/13758/images/flags/24/ru.png \"По-русски\")](https://codeforces.com/blog/entry/155813?locale=ru) \n\n[Enter](https://codeforces.com/enter?back=%2Fblog%2Fentry%2F155813)| [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* 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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* [Down\\_bad\\_4\\_haiender288](https://codeforces.com/profile/Down%5Fbad%5F4%5Fhaiender288 \"Pupil Down_bad_4_haiender288\") → [new manmade horrors (barely within comprehension) just dropped](https://codeforces.com/blog/entry/156826) ![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* [AlirezaBest](https://codeforces.com/profile/AlirezaBest \"Pupil AlirezaBest\") → [Is this thing regular?](https://codeforces.com/blog/entry/156867) ![New 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updated\")\n* [UNDERTAKER](https://codeforces.com/profile/UNDERTAKER \"Grandmaster UNDERTAKER\") → [Looking for contests to host — we'll adapt our judge to your problems](https://codeforces.com/blog/entry/156823) ![Text created or updated](//codeforces.com/codeforces.org/s/13758/images/icons/x-update-12x12.png \"Text created or updated\")\n* [Ovi\\_404](https://codeforces.com/profile/Ovi%5F404 \"Pupil Ovi_404\") → [Finally hit 1000 problems solved on Codeforces](https://codeforces.com/blog/entry/156830) ![New comment\\(s\\)](//codeforces.com/codeforces.org/s/13758/images/icons/comment-12x12.png \"New comment(s)\")\n* [yse](https://codeforces.com/profile/yse \"Candidate Master yse\") → [Codeforces Round 1114 (Div. 3) — Editorial](https://codeforces.com/blog/entry/155666) ![New comment\\(s\\)](//codeforces.com/codeforces.org/s/13758/images/icons/comment-12x12.png \"New comment(s)\")\n* [M.Bemanali](https://codeforces.com/profile/M.Bemanali \"Newbie M.Bemanali\") → [QueueForces AGAIN???](https://codeforces.com/blog/entry/156822) ![Text created or updated](//codeforces.com/codeforces.org/s/13758/images/icons/x-update-12x12.png \"Text created or updated\")\n* [BitByBit123](https://codeforces.com/profile/BitByBit123 \"Master BitByBit123\") → [Suspicious Master yql0991666](https://codeforces.com/blog/entry/155862) ![Text created or updated](//codeforces.com/codeforces.org/s/13758/images/icons/x-update-12x12.png \"Text created or updated\")\n* [lumibons](https://codeforces.com/profile/lumibons \"International Grandmaster lumibons\") → [IOI 2027 Call for Tasks](https://codeforces.com/blog/entry/156793) ![Text created or updated](//codeforces.com/codeforces.org/s/13758/images/icons/x-update-12x12.png \"Text created or updated\")\n* [Grizoo](https://codeforces.com/profile/Grizoo \"Candidate Master Grizoo\") → [\\[GYM\\] Aleppo Collegiate Programming Contest 2026](https://codeforces.com/blog/entry/156707) ![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\") → [Rule Restricting the Use of AI \\[revision 2024-09-14\\]](https://codeforces.com/blog/entry/133941) ![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* [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* [borgar02](https://codeforces.com/profile/borgar02 \"Master borgar02\") → [IATI 2022 editorial](https://codeforces.com/blog/entry/109545) ![New comment\\(s\\)](//codeforces.com/codeforces.org/s/13758/images/icons/comment-12x12.png \"New comment(s)\") ![Necropost](//codeforces.com/codeforces.org/s/13758/images/icons/hourglass.png \"Necropost\")\n* [Kerolloz](https://codeforces.com/profile/Kerolloz \"Newbie Kerolloz\") → [Codeforces JSON-Problem-Scraper API](https://codeforces.com/blog/entry/75074) ![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\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 193 — Editorial ](https://codeforces.com/blog/entry/155813) \n\nBy [BledDest](https://codeforces.com/profile/BledDest \"International Grandmaster BledDest\"), [history](https://codeforces.com/topic/156455/en1), 6 weeks ago, translation, ![In English](//codeforces.com/codeforces.org/s/13758/images/flags/24/gb.png \"In English\") \n\n[2253A - The Best Card](https://codeforces.com/contest/2253/problem/A \"Educational Codeforces Round 193 (Rated for Div. 2)\")\n\nIdea: [BledDest](https://codeforces.com/profile/BledDest \"International Grandmaster BledDest\")\n\n**Tutorial**\n\nTutorial is loading...\n\n**Solution (BledDest)**\n\n```\nt = int(input())\nfor i in range(t):\n    n = int(input())\n    good = True\n    for j in range(2, n + 1):\n        if (n + 1) % j == 0:\n            good = False\n    if good:\n        print('YES')\n    else:\n        print('NO')  \n\n\n```\n\n[2253B - Hypercarp and the Control Panel](https://codeforces.com/contest/2253/problem/B \"Educational Codeforces Round 193 (Rated for Div. 2)\")\n\nIdea: [FelixArg](https://codeforces.com/profile/FelixArg \"Master FelixArg\")\n\n**Tutorial**\n\nTutorial is loading...\n\n**Solution (FelixArg)**\n\n```\n#include <bits/stdc++.h>\n \nusing namespace std;\n#define int long long\n\nvoid solve(){\n\tint n;\n\tcin >> n;\n\tvector<int> a(n);\n\tfor (int i = 0; i < n; i++){\n\t\tcin >> a[i];\n\t}\n\tvector<pair<int, int>> b;\n\tfor (int i = 0; i < n; i++){\n\t\tif (b.empty() || b.back().first != a[i]){\n\t\t\tb.emplace_back(a[i], 1);\n\t\t}\n\t\telse{\n\t\t\tb.back().second++;\n\t\t}\n\t}\n\tint m = b.size();\n\tfor (int i = 0; i < m - 1; i++){\n\t\tif (b[i].second > 1 && b[i + 1].second > 1){\n\t\t\tcout << m + 2 << '\\n';\n\t\t\treturn;\n\t\t}\n\t}\n\tfor (int i = 0; i < m; i++){\n\t\tif (i < m - 1 && b[i].second > 1 && (i + 2 >= m || b[i + 2].first != b[i].first)){\n\t\t\tcout << m + 1 << '\\n';\n\t\t\treturn;\n\t\t}\n\t\tif (i > 0 && b[i].second > 1 && (i - 2 < 0 || b[i - 2].first != b[i].first)){\n\t\t\tcout << m + 1 << '\\n';\n\t\t\treturn;\n\t\t}\n\t}\n\tcout << m << '\\n';\n}\n\nsigned main()\n{\n#ifdef FELIX\n\tauto _clock_start = chrono::high_resolution_clock::now();\n#endif\n\tios_base::sync_with_stdio(false);\n\tcin.tie(0);\n\tcout.tie(0);\n \n\tint tests = 1;\n\tcin >> tests;\n\twhile(tests--){\n\t\tsolve();\n\t}\n \n#ifdef FELIX\n\tcerr << \"Executed in \" << chrono::duration_cast<chrono::milliseconds>(\n\t\tchrono::high_resolution_clock::now()\n\t\t\t- _clock_start).count() << \"ms.\" << endl;\n#endif\n\treturn 0;\n}\n \n\n```\n\n[2253C - Sum of Distinct Values in a Matrix](https://codeforces.com/contest/2253/problem/C \"Educational Codeforces Round 193 (Rated for Div. 2)\")\n\nIdea: [BledDest](https://codeforces.com/profile/BledDest \"International Grandmaster BledDest\")\n\n**Tutorial**\n\nTutorial is loading...\n\n**Solution (BledDest)**\n\n```\ndef get_k_last(l, k):\n    if k < len(l):\n        return l[len(l)-k:]\n    else:\n        return l\n\nt = int(input())\nfor _ in range(t):\n    n, m, x, y = map(int, input().split())\n    a = list(map(int, input().split()))\n    b = list(map(int, input().split()))\n    \n    c = []\n    d = []\n    e = []\n    i = 0\n    j = 0\n    while i < x and j < y:\n        if a[i] == b[j]:\n            c.append(a[i])\n            i += 1\n            j += 1\n        elif a[i] < b[j]:\n            d.append(a[i])\n            i += 1\n        else:\n            e.append(b[j])\n            j += 1\n    d.extend(a[i:])\n    e.extend(b[j:])\n    \n    res = sorted(get_k_last(d, n) + get_k_last(e, m) + c)\n    res = get_k_last(res, n + m - 1)\n    print(sum(res))\n\n\n```\n\n[2253D - Hypercarp and Interdimensional Jumps](https://codeforces.com/contest/2253/problem/D \"Educational Codeforces Round 193 (Rated for Div. 2)\")\n\nIdea: [FelixArg](https://codeforces.com/profile/FelixArg \"Master FelixArg\")\n\n**Tutorial**\n\nTutorial is loading...\n\n**Solution (FelixArg)**\n\n```\n#include <bits/stdc++.h>\n \nusing namespace std;\n#define int long long\n\nvoid solve(){\n\tint x, y;\n\tcin >> x >> y;\n\n\tint su = x + y;\n\tint k = 0;\n\twhile((k + 1) * (k + 2) / 2 <= su){\n\t\tk++;\n\t}\n\n\tint p1x = k * (k + 1) / 2 - y;\n\tint p1y = y;\n\n\tint p2x = x;\n\tint p2y = k * (k + 1) / 2 - x;\n\n\tvector<pair<int, int>> cand;\n\tif ((p1x + p2x) < 0){\n\t\tcand.emplace_back((p1x + p2x) / 2, k * (k + 1) / 2 - (p1x + p2x) / 2);\n\t\tcand.emplace_back((p1x + p2x - 1) / 2, k * (k + 1) / 2 - (p1x + p2x - 1) / 2);\n\t}\n\telse{\n\t\tcand.emplace_back((p1x + p2x) / 2, k * (k + 1) / 2 - (p1x + p2x) / 2);\n\t\tcand.emplace_back((p1x + p2x + 1) / 2, k * (k + 1) / 2 - (p1x + p2x + 1) / 2);\n\t}\n\n\tauto best = cand[0];\n\tfor (auto [p, q] : cand){\n\t\tif (p < 0){\n\t\t\tp = 0;\n\t\t}\n\t\tif (q < 0){\n\t\t\tq = 0;\n\t\t}\n\t\tif ((p - x) * (p - x) + (q - y) * (q - y) < \n\t\t\t(best.first - x) * (best.first - x) + (best.second - y) * (best.second - y)){\n\t\t\t\tbest = {p, q};\n\t\t}\n\t}\n\n\tstring ans(k, 'Y');\n\tfor (int i = 0; i < k; i++){\n\t\tif (best.first >= k - i){\n\t\t\tans[i] = 'X';\n\t\t\tbest.first -= (k - i);\n\t\t}\n\t}\n\n\tcout << ans << '\\n';\n}\n\nsigned main()\n{\n#ifdef FELIX\n\tauto _clock_start = chrono::high_resolution_clock::now();\n#endif\n\tios_base::sync_with_stdio(false);\n\tcin.tie(0);\n\tcout.tie(0);\n \n\tint tests = 1;\n\tcin >> tests;\n\twhile(tests--){\n\t\tsolve();\n\t}\n \n#ifdef FELIX\n\tcerr << \"Executed in \" << chrono::duration_cast<chrono::milliseconds>(\n\t\tchrono::high_resolution_clock::now()\n\t\t\t- _clock_start).count() << \"ms.\" << endl;\n#endif\n\treturn 0;\n}\n\n\n```\n\n[2253E - Diameter Intersections](https://codeforces.com/contest/2253/problem/E \"Educational Codeforces Round 193 (Rated for Div. 2)\")\n\nIdea: [BledDest](https://codeforces.com/profile/BledDest \"International Grandmaster BledDest\")\n\n**Tutorial**\n\nTutorial is loading...\n\n**Solution (BledDest)**\n\n```\n#include<bits/stdc++.h>\n\nusing namespace std;\n\nconst int N = 1000043;\n\nvector<int> g[N];\nint n;\n\nvector<int> get_dist(int x)\n{\n    vector<int> d(n, -1);\n    d[x] = 0;\n    queue<int> q;\n    q.push(x);\n    while(!q.empty())\n    {\n        int k = q.front();\n        q.pop();\n        for(auto y : g[k])\n            if(d[y] == -1)\n            {\n                d[y] = d[k] + 1;\n                q.push(y);\n            }   \n    }\n    return d;\n}\n\nvoid remove_edge(int x, int y)\n{\n    int idx = -1;\n    for(int i = 0; i < g[x].size(); i++)\n        if(g[x][i] == y)\n            idx = i;\n    g[x].erase(g[x].begin() + idx, g[x].begin() + idx + 1);    \n} \n\nvector<int> process(int v)\n{\n    vector<int> d(n, -1), p(n, -1);\n    queue<int> q;\n    q.push(v);\n    d[v] = 0;\n    vector<int> visited;\n    while(!q.empty())\n    {\n        int k = q.front();\n        q.pop();\n        visited.push_back(k);\n        for(auto y : g[k])\n            if(d[y] == -1)\n            {\n                d[y] = d[k] + 1;\n                q.push(y);\n                p[y] = k;\n            }\n    }\n    int max_dist = *max_element(d.begin(), d.end());\n    vector<bool> has_end(n, false);\n    has_end[v] = true;\n    for(auto x : visited)\n        if(d[x] == max_dist)\n        {\n            int cur = x;\n            while(!has_end[cur])\n            {\n                has_end[cur] = true;\n                cur = p[cur];\n            }\n        }\n\n    vector<int> res;\n    for(auto x : visited)\n    {\n        if(!has_end[x]) continue;\n        int good_children = 0;\n        for(auto y : g[x])\n            if(p[x] != y && has_end[y]) good_children++;\n        if(good_children != 1) res.push_back(d[x]);\n    }   \n    sort(res.begin(), res.end());\n    res.erase(unique(res.begin(), res.end()), res.end());\n    return res;\n}\n\nvoid solve()\n{\n    cin >> n;\n    for(int i = 0; i < n; i++)\n        g[i].clear();\n    for(int i = 0; i < n - 1; i++)\n    {\n        int x, y;\n        cin >> x >> y;\n        --x;\n        --y;\n        g[x].push_back(y);\n        g[y].push_back(x);\n    }\n\n    auto dist0 = get_dist(0);\n    int e1 = max_element(dist0.begin(), dist0.end()) - dist0.begin();\n    auto dist1 = get_dist(e1);\n    int e2 = max_element(dist1.begin(), dist1.end()) - dist1.begin();\n    auto dist2 = get_dist(e2);\n    int d = dist1[e2];\n\n    int x = -1, y = -1;\n    for(int i = 0; i < n; i++)\n        if(dist1[i] + dist2[i] == d)\n        {\n            if(dist1[i] == dist2[i] - 1)\n                x = i;\n            else if(dist1[i] == dist2[i] + 1)\n                y = i;\n        }\n\n    remove_edge(x, y);\n    remove_edge(y, x);\n    auto ans1 = process(x);\n    auto ans2 = process(y);\n\n    vector<bool> res(n + 1);\n    for(auto x : ans1)\n        for(auto y : ans2)\n            res[x + y + 1] = true;\n    int cnt = 0;\n    for(auto x : res)\n        if(x) cnt++;\n    cout << cnt;\n    for(int i = 0; i <= n; i++)\n        if(res[i])\n            cout << \" \" << i;\n    cout << 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[2253F - 4-beauty](https://codeforces.com/contest/2253/problem/F \"Educational Codeforces Round 193 (Rated for Div. 2)\")\n\nIdea: [BledDest](https://codeforces.com/profile/BledDest \"International Grandmaster BledDest\")\n\n**Tutorial**\n\nTutorial is loading...\n\n**Solution (BledDest)**\n\n```\n#include<bits/stdc++.h>\n\nusing namespace std;\n\nconst int N = int(5e5) + 43;\nconst long long INF64 = (long long)(1e18);\n\nint n;\nint cost[N];\n\nvoid upd(long long& x, long long y)\n{\n    if(x > y) x = y;\n}\n\nbool check_bit(int x, int y)\n{\n    return bool((x >> y) & 1);\n}\n\nlong long calc_dp(const vector<vector<int>>& c)\n{\n    int n = c.size();\n    int m = c[0].size();\n    int full = (1 << m) - 1;\n    vector<vector<vector<long long>>> dp(2, vector<vector<long long>>(m, vector<long long>(1 << m, INF64)));\n    dp[0][0][0] = 0;\n    for(int i = 0; i < n; i++)\n    {\n        int i1 = i & 1;\n        int i2 = i1 ^ 1;\n        for(int j = 0; j < m; j++)\n            for(int f = 0; f < (1 << m); f++)\n                dp[i2][j][f] = INF64;\n        for(int j = 0; j < m; j++)\n            for(int f = 0; f < (1 << m); f++)\n            {\n                if(dp[i1][j][f] == INF64) continue;\n                int ni = i1;\n                int nj = j + 1;\n                if(nj == m)\n                {\n                    ni = i2;\n                    nj = 0;\n                }\n                bool can = true;\n                if(i > 0 && j + 2 < m && check_bit(f, m - 1) && check_bit(f, m - 2) && check_bit(f, m - 3))\n                    can = false;\n                if(can)\n                {\n                    int nf = ((f << 1) & full) | 1;\n                    upd(dp[ni][nj][nf], dp[i1][j][f]); \n                }\n                int nf = (f << 1) & full;\n                upd(dp[ni][nj][nf], dp[i1][j][f] + c[i][j]);\n            }\n    }   \n    return *min_element(dp[n & 1][0].begin(), dp[n & 1][0].end());\n}\n\nlong long calc_comp(int x)\n{\n    int d2 = 0, d3 = 0;\n    int p2 = 1, p3 = 1;\n    while(x * (p2 * 2) <= n)\n    {\n        p2 *= 2;\n        d2++;\n    }\n    while(x * (p3 * 3) <= n)\n    {\n        p3 *= 3;\n        d3++;\n    }\n    d2++;\n    d3++;\n    vector<int> pow2(d2, 1), pow3(d3, 1);\n    for(int i = 1; i < d2; i++)\n        pow2[i] = pow2[i - 1] * 2;\n    for(int i = 1; i < d3; i++)\n        pow3[i] = pow3[i - 1] * 3;\n    vector<vector<int>> cur(d3, vector<int>(d2, 0));\n    for(int i = 0; i < d2; i++)\n        for(int j = 0; j < d3; j++)\n            if(x * 1ll * pow2[i] * 1ll * pow3[j] <= n)\n                cur[j][i] = cost[x * pow2[i] * pow3[j]];\n    return calc_dp(cur);\n}   \n\nvoid solve()\n{\n    cin >> n;\n    for(int i = 1; i <= n; i++)\n        cin >> cost[i];\n    long long ans = 0;\n    for(int i = 1; i <= n; i++)\n        if(i % 2 != 0 && i % 3 != 0)\n            ans += calc_comp(i);\n    cout << ans << endl; \n}\n\nint main()\n{\n    ios_base::sync_with_stdio(0);\n    int t = 1;\n    for(int i = 0; i < t; i++)\n        solve();    \n}  \n\n\n```\n\n![](//codeforces.com/codeforces.org/s/13758/images/icons/paperclip-16x16.png) Tutorial of [Educational Codeforces Round 193 (Rated for Div. 2)](https://codeforces.com/contest/2253) \n\n* [![Vote: I like it](//codeforces.com/codeforces.org/s/13758/images/actions/voteup-gray.png \"Vote: I like it\")](#)\n* +55\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\") 6 weeks ago\n* [![Comments](//codeforces.com/codeforces.org/s/13758/images/blog/comments_16x16.png \"Comments\")](https://codeforces.com/blog/entry/155813#comments) [22 ](https://codeforces.com/blog/entry/155813#comments)\n\n  \n![Comments](//codeforces.com/codeforces.org/s/13758/images/icons/comments-48x48.png \"Comments\") Comments (22) \n\n[Write comment?](#) \n\n| » [salemi1](https://codeforces.com/profile/salemi1 \"Specialist salemi1\") | 6 weeks ago, [show (+1)](#) # |\n| ------------------------------------------------------------------------ | ----------------------------- |\n\n| » [ ![](https://codeforces.com/userpic.codeforces.org/4817146/avatar/2167beab4eeac4e9.jpg) ](https://codeforces.com/profile/salemi1) [salemi1](https://codeforces.com/profile/salemi1 \"Specialist salemi1\") | 6 weeks ago, [hide](#) [#](https://codeforces.com/blog/entry/155813?#comment-1384199 \"Link to comment\") \\| [←](#)Rev. 2 [→](#) [![Vote: I like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) +3 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) But D just needed a simple greedy [386119523](https://codeforces.com/contest/2253/submission/386119523 \"Submission 386119523 by salemi1\") **The code**void solve() {     int n,m; cin>>n>>m;       int L=0, R=20010;     while(L+1!=R) {         int cen=(L+R)/2;         int a=n, b=m;                  for(int i=cen; i>0; i--) {             if(a>b) a-= i;             else b-= i;         }           if(a<0 or b<0) {             R=cen;         } else {             L=cen;         }     }       int a=n, b=m;     for(int i=L; i>0; i--) {         if(a>b) { a-=i; cout<<'X'; }         else { b-=i; cout<<'Y'; }     } cout<<endl;   } [→](#) [Reply](#) |\n| ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ |\n\n* | » » [Noobish\\_Monk](https://codeforces.com/profile/Noobish%5FMonk \"Master Noobish_Monk\") | 6 weeks ago, [show (+1)](#) # |\n| ---------------------------------------------------------------------------------------- | ----------------------------- |\n\n| » » [ ![](https://codeforces.com/userpic.codeforces.org/2080623/avatar/1db04b38e51c59b.jpg) ](https://codeforces.com/profile/Noobish%5FMonk) [Noobish\\_Monk](https://codeforces.com/profile/Noobish%5FMonk \"Master Noobish_Monk\") | 6 weeks ago, [hide](#) [#](https://codeforces.com/blog/entry/155813?#comment-1384215 \"Link to comment\") [^](#comment-1384199 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) +3 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) Mind explaining why greedy works? [→](#) [Reply](#) |\n| --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n  * | » » » [salemi1](https://codeforces.com/profile/salemi1 \"Specialist salemi1\") | 6 weeks ago, [show (+1)](#) # |\n| ---------------------------------------------------------------------------- | ----------------------------- |\n\n| » » » [ ![](https://codeforces.com/userpic.codeforces.org/4817146/avatar/2167beab4eeac4e9.jpg) ](https://codeforces.com/profile/salemi1) [salemi1](https://codeforces.com/profile/salemi1 \"Specialist salemi1\") | 6 weeks ago, [hide](#) [#](https://codeforces.com/blog/entry/155813?#comment-1384299 \"Link to comment\") [^](#comment-1384215 \"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\")](#) ( we move from (x,y) to (0,0) )First of all, we know the maximum operations we can perform(k). the 'i'-th change in 'a' or 'b' will finally reduce 'x' or 'y' by k+1-i if one of the numbers is much bigger and can't reach the other number by using all the operation, so it's OK!Otherwise we consider that x>y then, we have a moment that 'x' reachs 'y' and probably become smaller than 'y'. in this case, the maximum difference between 'x' and 'y' is 'k' so it compensates with k-1 (and k-1 with k-2...) and finally the maximum difference becomes 1And the minimum difference between x and y is what we need! [→](#) [Reply](#) |\n| --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ |\n\n    * | » » » » [KnownAsJason](https://codeforces.com/profile/KnownAsJason \"Expert KnownAsJason\") | 6 weeks ago, [show](#) # |\n| ----------------------------------------------------------------------------------------- | ------------------------ |\n\n| » » » » [ ![](https://codeforces.com/userpic.codeforces.org/4710940/avatar/a36a72563a88c277.jpg) ](https://codeforces.com/profile/KnownAsJason) [KnownAsJason](https://codeforces.com/profile/KnownAsJason \"Expert KnownAsJason\") | 6 weeks ago, [hide](#) [#](https://codeforces.com/blog/entry/155813?#comment-1384327 \"Link to comment\") [^](#comment-1384299 \"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\")](#) holy shoots this is brilliant [→](#) [Reply](#) |\n| --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n| » [Anush2010](https://codeforces.com/profile/Anush2010 \"Expert Anush2010\") | 6 weeks ago, [show](#) # |\n| -------------------------------------------------------------------------- | ------------------------ |\n\n| » [ ![](https://codeforces.com/userpic.codeforces.org/4268424/avatar/4ff1c1e778200912.jpg) ](https://codeforces.com/profile/Anush2010) [Anush2010](https://codeforces.com/profile/Anush2010 \"Expert Anush2010\") | 6 weeks ago, [hide](#) [#](https://codeforces.com/blog/entry/155813?#comment-1384207 \"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\")](#) Yeaah, finally ))) [→](#) [Reply](#) |\n| --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n| » [Misuki](https://codeforces.com/profile/Misuki \"Master Misuki\") | 6 weeks ago, [show](#) # |\n| ----------------------------------------------------------------- | ------------------------ |\n\n| » [ ![](https://codeforces.com/userpic.codeforces.org/1197572/avatar/e4ea7bc4debf5d23.jpg) ](https://codeforces.com/profile/Misuki) [Misuki](https://codeforces.com/profile/Misuki \"Master Misuki\") | 6 weeks ago, [hide](#) [#](https://codeforces.com/blog/entry/155813?#comment-1384211 \"Link to comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) +8 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) Alternative solution of D:Let $$$k$$$ be the largest integer with $$$\\\\frac{k \\\\cdot (k + 1)}{2} \\\\le x + y$$$. Consider two bag with capacity $$$x, y$$$ respectively. For $$$w = k, k - 1, \\\\ldots, 1$$$ in this order, put the item with weight $$$w$$$ into the bag with larger capacity remain. Let $$$x', y'$$$ denote the final capacity remain for two bags. We claim this process minimize $$$\\\\max(x', y')$$$.Proof.W.L.O.G. assume $$$x \\\\ge y$$$, call the bag with $$$x$$$ initial capacity the first bag, and the other be the second bag.case 1 ($$$x \\\\ge y + \\\\frac{k\\\\cdot (k + 1)}{2}$$$): trivialcase 2: items would be add into the first bag until its capacity is no more than the second bag, let $$$m$$$ be the item added to make this happen. then at this point, the difference of capacity between two bag would be no more than $$$m$$$, and in the subsequent item addition, the difference would be no more than the item added last, thus $$$|x' - y'| \\\\le 1$$$ holds at the end, which achieve the lower bound of $$$\\\\max(x', y')$$$. [→](#) [Reply](#) |\n| --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n| » [isagiyoichi11](https://codeforces.com/profile/isagiyoichi11 \"Specialist isagiyoichi11\") | 6 weeks ago, [show (+1)](#) # |\n| ------------------------------------------------------------------------------------------ | ----------------------------- |\n\n| » [ ![](https://codeforces.com/userpic.codeforces.org/5725147/avatar/c3a4b891570882b2.jpg) ](https://codeforces.com/profile/isagiyoichi11) [isagiyoichi11](https://codeforces.com/profile/isagiyoichi11 \"Specialist isagiyoichi11\") | 6 weeks ago, [hide](#) [#](https://codeforces.com/blog/entry/155813?#comment-1384212 \"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\")](#) In Problem E (2253E), how would i know number of distinct LCAs would be O($$$\\\\sqrt{n}$$$)? I was stuck with thinking it was O($$$n^2$$$) solution and won't pass. [→](#) [Reply](#) |\n| ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n* | » » [oelyas](https://codeforces.com/profile/oelyas \"Specialist oelyas\") | 6 weeks ago, [show](#) # |\n| ----------------------------------------------------------------------- | ------------------------ |\n\n| » » [ ![](https://codeforces.com/userpic.codeforces.org/4291105/avatar/f500eefdcb6fcd52.jpg) ](https://codeforces.com/profile/oelyas) [oelyas](https://codeforces.com/profile/oelyas \"Specialist oelyas\") | 6 weeks ago, [hide](#) [#](https://codeforces.com/blog/entry/155813?#comment-1384233 \"Link to comment\") [^](#comment-1384212 \"Parent comment\") \\| [←](#)Rev. 2 [→](#) [![Vote: I like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) We have $$$n-1$$$ edges, and each edge either part of diameter or not, and each node of diameter can form just a single distinct number, and sum of nodes limited on $$$n$$$, so you wont have more than $$$sqrt(n)$$$ distinct numbers. We can prove that by summing the first k small values such that sum won't exceed $$$n$$$. [→](#) [Reply](#) |\n| --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ |\n* | » » [mgranger22](https://codeforces.com/profile/mgranger22 \"Candidate Master mgranger22\") | 6 weeks ago, [show](#) # |\n| ----------------------------------------------------------------------------------------- | ------------------------ |\n\n| » » [ ![](https://codeforces.com/userpic.codeforces.org/3637499/avatar/c2e543d66773ecca.jpg) ](https://codeforces.com/profile/mgranger22) [mgranger22](https://codeforces.com/profile/mgranger22 \"Candidate Master mgranger22\") | 6 weeks ago, [hide](#) [#](https://codeforces.com/blog/entry/155813?#comment-1384245 \"Link to comment\") [^](#comment-1384212 \"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\")](#) Firstly discard all nodes who's subtree does NOT have a descendant of maximal depth, as these can never be valid LCA's. For the remaining nodes, define $$$cnt\\[d\\]$$$ as the number of nodes at depth $$$d$$$. Since each path is guaranteed to continue down until the maximal depth, we know that $$$cnt\\[d + 1\\] \\\\geq cnt\\[d\\]$$$. Furthermore, when depth $$$d$$$ contains a valid LCA (i.e. some node has two or more children with max depth descendants), then $$$cnt\\[d + 1\\] \\\\geq cnt\\[d\\] + 1$$$. If there are $$$r$$$ distinct achievable depths, then the width increases at least $$$r$$$ times. After the i-th increase, the width is at least $$$i+1$$$. Therefore, among the levels containing these increases and the level immediately after each one, there are at least $$$1 + 2 + \\\\ldots + (r + 1)$$$ nodes. Since the number of nodes per valid LCA depth grows quadratically, the number of valid LCA depths per node grows at a rate of $$$O(\\\\sqrt{n})$$$. [→](#) [Reply](#) |\n| ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n| » [mgranger22](https://codeforces.com/profile/mgranger22 \"Candidate Master mgranger22\") | 6 weeks ago, [show (+1)](#) # |\n| --------------------------------------------------------------------------------------- | ----------------------------- |\n\n| » [ ![](https://codeforces.com/userpic.codeforces.org/3637499/avatar/c2e543d66773ecca.jpg) ](https://codeforces.com/profile/mgranger22) [mgranger22](https://codeforces.com/profile/mgranger22 \"Candidate Master mgranger22\") | 6 weeks ago, [hide](#) [#](https://codeforces.com/blog/entry/155813?#comment-1384246 \"Link to comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/13758/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) +8 [![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 believe there's a typo in F. I think it should be x mod 2 != 0 and x mod 3 != 0 whereas it says the opposite now. [→](#) [Reply](#) |\n| ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n* | » » [BledDest](https://codeforces.com/profile/BledDest \"International Grandmaster BledDest\") | 6 weeks ago, [show](#) # |\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\") | 6 weeks ago, [hide](#) [#](https://codeforces.com/blog/entry/155813?#comment-1384328 \"Link to comment\") [^](#comment-1384246 \"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\")](#) Thank you, fixed that. The editorial should update in a couple of minutes [→](#) [Reply](#) |\n| ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n| » [LNight121](https://codeforces.com/profile/LNight121 \"Master LNight121\") | 6 weeks ago, [show](#) # |\n| -------------------------------------------------------------------------- | ------------------------ |\n\n\n...23195 bytes truncated...\n\nThe tool call succeeded but the output was truncated. Full output saved to: /Users/zayd/.local/share/opencode/tool-output/tool_0b2527a0e0018EIlArqNo2XNuZ\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.",
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              "command": "python3 -c 'from math import isqrt\nfor n in range(2, 301):\n    wins = lambda x, y: x < y if max(x, y) % min(x, y) == 0 else x > y\n    actual = any(all(x == y or wins(x, y) for y in range(2, n + 2)) for x in range(2, n + 2))\n    expected = all((n + 1) % d for d in range(2, isqrt(n + 1) + 1))\n    assert actual == expected, n\nprint(\"Brute-force comparison passed for n = 2 through 300.\")'"
            },
            "output": "Brute-force comparison passed for n = 2 through 300.\n",
            "metadata": {
              "output": "Brute-force comparison passed for n = 2 through 300.\n",
              "exit": 0,
              "truncated": false
            },
            "title": "python3 -c 'from math import isqrt\nfor n in range(2, 301):\n    wins = lambda x, y: x < y if max(x, y) % min(x, y) == 0 else x > y\n    actual = any(all(x == y or wins(x, y) for y in range(2, n + 2)) for x in range(2, n + 2))\n    expected = all((n + 1) % d for d in range(2, isqrt(n + 1) + 1))\n    assert actual == expected, n\nprint(\"Brute-force comparison passed for n = 2 through 300.\")'",
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    {
      "info": {
        "parentID": "msg_0b25267f9001dKimfBE2zpQytf",
        "role": "assistant",
        "mode": "nudge-generation",
        "agent": "nudge-generation",
        "variant": "low",
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          "reason": null,
          "hints": [
            {
              "order": 1,
              "content": "You don't need to compare every pair of cards. Try finding just one opponent that rules out a candidate."
            },
            {
              "order": 2,
              "content": "For a card with value $x<n+1$, the card $x+1$ also exists. Can $x$ divide $x+1$ when $x\\ge 2$? Which card wins?"
            },
            {
              "order": 3,
              "content": "Every card except $n+1$ loses to the next card. So $n+1$ is the only possible answer. What kind of smaller card could beat it?"
            },
            {
              "order": 4,
              "content": "A smaller card $d$ beats $n+1$ exactly when $d$ divides $n+1$. Therefore, a card that beats everyone exists exactly when $n+1$ is prime."
            },
            {
              "order": 5,
              "content": "Test $m=n+1$ for primality by checking divisors from $2$ while $d^2\\le m$. Every composite number has a divisor in that range. If you find one, print `NO`; otherwise, print `YES`. This takes $O(\\sqrt{n+1})$ time and $O(1)$ extra space per test case."
            }
          ],
          "editorial": "The largest card isn't automatically the strongest—divisibility can ruin its day. But it turns out to be the **only possible card that beats everyone**.\n\n### 1. Eliminate every smaller card\n\nTake any card $x<n+1$. Since the deck contains every integer from $2$ through $n+1$, the card $x+1$ exists too.\n\nNeither number divides the other:\n\n- $x+1$ cannot divide the smaller positive number $x$.\n- If $x$ divided $x+1$, it would also divide $(x+1)-x=1$. That's impossible because $x\\ge 2$.\n\nSo the larger card, $x+1$, wins. This rules out **every card except $n+1$**.\n\n### 2. Check the only remaining candidate\n\nLet $m=n+1$. Every other card has a value $d$ with $2\\le d<m$.\n\n- If $d$ divides $m$, the smaller card $d$ wins.\n- Otherwise, neither value divides the other, so the larger card $m$ wins.\n\nThus, $m$ beats every other card exactly when it has no divisor between $2$ and $m-1$. That's precisely the definition of a prime number.\n\nTherefore,\n\n$$\\boxed{\\text{Print YES if and only if }n+1\\text{ is prime.}}$$\n\nCareful with the off-by-one: $n$ is the **number of cards**, while $n+1$ is the largest card value.\n\n### Algorithm\n\nFor each test case:\n\n1. Set $m=n+1$.\n2. Try each integer divisor $d$ starting at $2$, while $d^2\\le m$.\n3. If any divisor satisfies $m\\bmod d=0$, print `NO`.\n4. If no divisor is found, print `YES`.\n\nWhy stop at the square root? If $m=ab$ is composite and both factors were greater than $\\sqrt m$, their product would exceed $m$. So at least one nontrivial divisor is at most $\\sqrt m$.\n\nUse an inclusive condition, `d * d <= m`, so perfect squares such as $9$ are correctly rejected. The code uses `1LL * d * d` to evaluate the product as a 64-bit integer.\n\n### Correctness proof\n\nEvery card $x<n+1$ loses to the card $x+1$, so no such card can beat everyone. Only $m=n+1$ can qualify.\n\nIf $m$ is composite, it has a divisor $d$ with $2\\le d<m$. That divisor is present in the deck and beats $m$, so no qualifying card exists.\n\nIf $m$ is prime, no other card divides it. Since every other card is smaller than $m$, the larger-value rule makes $m$ win against all of them. A qualifying card therefore exists.\n\nThe divisor loop correctly determines whether $m$ is prime, so the algorithm prints `YES` exactly when the required card exists.\n\n### Complexity\n\nEach test case takes $O(\\sqrt{n+1})$ time and $O(1)$ extra space. With $n+1\\le 200001$, there are fewer than $450$ candidate divisors per test case.",
          "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\nint main() {\n    setIO();\n\n    int t;\n    cin >> t;\n    while (t--) {\n        int n;\n        cin >> n;\n        int m = n + 1;\n        bool prime = true;\n        for (int d = 2; 1LL * d * d <= m; ++d) {\n            if (m % d == 0) {\n                prime = false;\n                break;\n            }\n        }\n        cout << (prime ? \"YES\\n\" : \"NO\\n\");\n    }\n}\n"
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          "type": "tool",
          "tool": "StructuredOutput",
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          "state": {
            "status": "completed",
            "input": {
              "status": "success",
              "reason": null,
              "hints": [
                {
                  "order": 1,
                  "content": "You don't need to compare every pair of cards. Try finding just one opponent that rules out a candidate."
                },
                {
                  "order": 2,
                  "content": "For a card with value $x<n+1$, the card $x+1$ also exists. Can $x$ divide $x+1$ when $x\\ge 2$? Which card wins?"
                },
                {
                  "order": 3,
                  "content": "Every card except $n+1$ loses to the next card. So $n+1$ is the only possible answer. What kind of smaller card could beat it?"
                },
                {
                  "order": 4,
                  "content": "A smaller card $d$ beats $n+1$ exactly when $d$ divides $n+1$. Therefore, a card that beats everyone exists exactly when $n+1$ is prime."
                },
                {
                  "order": 5,
                  "content": "Test $m=n+1$ for primality by checking divisors from $2$ while $d^2\\le m$. Every composite number has a divisor in that range. If you find one, print `NO`; otherwise, print `YES`. This takes $O(\\sqrt{n+1})$ time and $O(1)$ extra space per test case."
                }
              ],
              "editorial": "The largest card isn't automatically the strongest—divisibility can ruin its day. But it turns out to be the **only possible card that beats everyone**.\n\n### 1. Eliminate every smaller card\n\nTake any card $x<n+1$. Since the deck contains every integer from $2$ through $n+1$, the card $x+1$ exists too.\n\nNeither number divides the other:\n\n- $x+1$ cannot divide the smaller positive number $x$.\n- If $x$ divided $x+1$, it would also divide $(x+1)-x=1$. That's impossible because $x\\ge 2$.\n\nSo the larger card, $x+1$, wins. This rules out **every card except $n+1$**.\n\n### 2. Check the only remaining candidate\n\nLet $m=n+1$. Every other card has a value $d$ with $2\\le d<m$.\n\n- If $d$ divides $m$, the smaller card $d$ wins.\n- Otherwise, neither value divides the other, so the larger card $m$ wins.\n\nThus, $m$ beats every other card exactly when it has no divisor between $2$ and $m-1$. That's precisely the definition of a prime number.\n\nTherefore,\n\n$$\\boxed{\\text{Print YES if and only if }n+1\\text{ is prime.}}$$\n\nCareful with the off-by-one: $n$ is the **number of cards**, while $n+1$ is the largest card value.\n\n### Algorithm\n\nFor each test case:\n\n1. Set $m=n+1$.\n2. Try each integer divisor $d$ starting at $2$, while $d^2\\le m$.\n3. If any divisor satisfies $m\\bmod d=0$, print `NO`.\n4. If no divisor is found, print `YES`.\n\nWhy stop at the square root? If $m=ab$ is composite and both factors were greater than $\\sqrt m$, their product would exceed $m$. So at least one nontrivial divisor is at most $\\sqrt m$.\n\nUse an inclusive condition, `d * d <= m`, so perfect squares such as $9$ are correctly rejected. The code uses `1LL * d * d` to evaluate the product as a 64-bit integer.\n\n### Correctness proof\n\nEvery card $x<n+1$ loses to the card $x+1$, so no such card can beat everyone. Only $m=n+1$ can qualify.\n\nIf $m$ is composite, it has a divisor $d$ with $2\\le d<m$. That divisor is present in the deck and beats $m$, so no qualifying card exists.\n\nIf $m$ is prime, no other card divides it. Since every other card is smaller than $m$, the larger-value rule makes $m$ win against all of them. A qualifying card therefore exists.\n\nThe divisor loop correctly determines whether $m$ is prime, so the algorithm prints `YES` exactly when the required card exists.\n\n### Complexity\n\nEach test case takes $O(\\sqrt{n+1})$ time and $O(1)$ extra space. With $n+1\\le 200001$, there are fewer than $450$ candidate divisors per test case.",
              "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\nint main() {\n    setIO();\n\n    int t;\n    cin >> t;\n    while (t--) {\n        int n;\n        cin >> n;\n        int m = n + 1;\n        bool prime = true;\n        for (int d = 2; 1LL * d * d <= m; ++d) {\n            if (m % d == 0) {\n                prime = false;\n                break;\n            }\n        }\n        cout << (prime ? \"YES\\n\" : \"NO\\n\");\n    }\n}\n"
            },
            "output": "Structured output captured successfully.",
            "metadata": {
              "valid": true
            },
            "title": "Structured Output",
            "time": {
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              "end": 1789698192570
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