{ "info": { "id": "ses_f520d9080ffeOzR9O7LuPqvgeA", "slug": "calm-orchid", "projectID": "global", "directory": "/private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-QYeIIC", "path": "private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-QYeIIC", "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": 22341, "output": 1371, "reasoning": 25, "cache": { "read": 8832, "write": 0 } }, "time": { "created": 1789624741759, "updated": 1789624797288 } }, "messages": [ { "info": { "role": "user", "time": { "created": 1789624741777 }, "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_0adf26f91001j25fyMcm8mcDRY", "sessionID": "ses_f520d9080ffeOzR9O7LuPqvgeA" }, "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 2259D: \"MEX Multiset\" (rated 1200).\n\nKnown problem metadata:\n- Contest ID: 2259\n- Index: D\n- Name: MEX Multiset\n- Rating: 1200 (Codeforces difficulty rating)\n- Tags: constructive algorithms, greedy\n- Problem URL: https://codeforces.com/contest/2259/problem/D\n\nUse the rating and tags as weak signals only. The supplied statement is the source of truth.\n\nProblem Statement:\n\n
D. MEX Multiset
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

You are given an array $$$a_1, a_2, \\ldots, a_n$$$. There exist $$$3$$$ initially empty multisets $$$A, B, C$$$, and for each index $$$i$$$ ($$$1 \\leq i \\leq n$$$), you may put $$$a_i$$$ into exactly one of $$$A$$$, $$$B$$$, or $$$C$$$.

Determine whether it is possible to put the elements into the multisets such that $$$\\operatorname{MEX}(A) + \\operatorname{MEX}(B) + \\operatorname{MEX}(C) \\geq 2 \\cdot \\max(\\operatorname{MEX}(A), \\operatorname{MEX}(B), \\operatorname{MEX}(C))$$$$$$^{\\text{∗}}$$$. If so, output a construction that achieves this.

$$$^{\\text{∗}}$$$$$$\\operatorname{MEX}(D)$$$ is defined as the smallest non-negative integer that is not present in the set $$$D$$$. For example, $$$\\operatorname{MEX}([1, 2, 0, 5]) = 3$$$, and $$$\\operatorname{MEX}([1, 2, 4, 9]) = 0$$$. The $$$\\operatorname{MEX}$$$ of an empty set is $$$0$$$.

Input

The first line of each input contains $$$t$$$ ($$$1 \\leq t \\leq 10^4$$$) — the number of test cases.

The first line of each test case contains $$$n$$$ ($$$3 \\leq n \\leq 2 \\cdot 10^5$$$) — the length of $$$a$$$.

The second line of each test case contains $$$a_1, a_2, \\ldots, a_n$$$ ($$$0 \\leq a_i \\leq 10^9$$$) — the array $$$a$$$.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \\cdot 10^5$$$.

Output

If a valid distribution of elements into the multisets exists, output $$$\\texttt{YES}$$$. Otherwise, output $$$\\texttt{NO}$$$.

If the answer is $$$\\texttt{YES}$$$, output a string $$$s$$$ of length $$$n$$$ on a new line, such that $$$s_i = \\texttt{A}$$$ if the $$$i$$$-th element was put into the multiset $$$A$$$, $$$s_i = \\texttt{B}$$$ if the $$$i$$$-th element was put into the multiset $$$B$$$, and $$$s_i = \\texttt{C}$$$ if the $$$i$$$-th element was put into the multiset $$$C$$$.

You can output the answer in any case (upper or lower). For example, the strings $$$\\texttt{YES}$$$, $$$\\texttt{yes}$$$, $$$\\texttt{yEs}$$$, and $$$\\texttt{Yes}$$$ will be recognized as positive responses, and the strings $$$\\texttt{NO}$$$, $$$\\texttt{no}$$$, $$$\\texttt{No}$$$ will be recognized as negative responses. Additionally, the strings $$$\\texttt{AABCAAA}$$$, $$$\\texttt{aabcaaa}$$$, and $$$\\texttt{aaBcaaa}$$$ will be recognized as the same answer.

If there are multiple possible outputs, output any.

Example
Input
5
6
1 0 0 1 2 1
4
0 0 0 0
3
0 2 2
4
6 7 6 7
5
0 0 0 1 2
Output
YES\nABABCA\nYES\nABAC\nNO\nYES\nAAAB\nYES\nABCAB\n
Note

In the first test case, we can have $$$A = \\{0, 1, 1\\}$$$, $$$B = \\{0, 1\\}$$$, $$$C = \\{2\\}$$$, meaning $$$\\operatorname{MEX}(A) + \\operatorname{MEX}(B) + \\operatorname{MEX}(C) = 4$$$, and $$$2 \\cdot \\max(\\operatorname{MEX}(A), \\operatorname{MEX}(B), \\operatorname{MEX}(C)) = 2 \\cdot \\max(2, 2, 0) = 4$$$.

In the third test case, it can be shown that there are no valid distributions.

\n\n\n\nSource lookup status:\n\n- Tutorial (en): https://codeforces.com/blog/entry/156457 loaded (200 OK; title: \"Codeforces Round 1119 (Div. 3) Editorial - Codeforces\"; mentions 2259D)\n\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 \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_0adf26f92001L7V3NkmGbYb5ZO", "sessionID": "ses_f520d9080ffeOzR9O7LuPqvgeA", "messageID": "msg_0adf26f91001j25fyMcm8mcDRY" } ] }, { "info": { "parentID": "msg_0adf26f91001j25fyMcm8mcDRY", "role": "assistant", "mode": "nudge-generation", "agent": "nudge-generation", "variant": "low", "path": { "cwd": "/private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-QYeIIC", "root": "/" }, "cost": 0, "tokens": { "total": 9152, "input": 9120, "output": 32, "reasoning": 0, "cache": { "write": 0, "read": 0 } }, "modelID": "gpt-6-astra", "providerID": "openai", "time": { "created": 1789624741906, "completed": 1789624751750 }, "finish": "tool-calls", "id": "msg_0adf270120010LiuZDmVLbToj2", "sessionID": "ses_f520d9080ffeOzR9O7LuPqvgeA" }, "parts": [ { "type": "step-start", "id": "prt_0adf27d01001vlA9nm7M6ziF2L", "sessionID": "ses_f520d9080ffeOzR9O7LuPqvgeA", "messageID": "msg_0adf270120010LiuZDmVLbToj2" }, { "type": "tool", "tool": "webfetch", "callID": "call_0KzXuoEtYN9nFT2kuPmkrQPm", "state": { "status": "completed", "input": { "url": "https://codeforces.com/blog/entry/156457", "format": "markdown" }, "output": "---\nimage: https://codeforces.com/codeforces.org/s/85295/images/codeforces-telegram-square-1024x1024.png\ntitle: Codeforces Round 1119 (Div. 3) Editorial - Codeforces\ndescription: Codeforces. Programming competitions and contests, programming community\n---\n\n \n\n \n[![Codeforces](//codeforces.com/codeforces.org/s/85295/images/codeforces-logo-with-telegram-555x130.png \"Codeforces\")](https://codeforces.com/) \n\n[![In English](//codeforces.com/codeforces.org/s/85295/images/flags/24/gb.png \"In English\")](https://codeforces.com/blog/entry/156457?locale=en) [![По-русски](//codeforces.com/codeforces.org/s/85295/images/flags/24/ru.png \"По-русски\")](https://codeforces.com/blog/entry/156457?locale=ru) \n\n[Enter](https://codeforces.com/enter?back=%2Fblog%2Fentry%2F156457)| [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* 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![Text created or updated](//codeforces.com/codeforces.org/s/85295/images/icons/x-update-12x12.png \"Text created or updated\")\n* [kondasujay2](https://codeforces.com/profile/kondasujay2 \"International Master kondasujay2\") → [Codeforces Round 1120 (Div 1, Div 2) Editorial](https://codeforces.com/blog/entry/156688) ![Text created or updated](//codeforces.com/codeforces.org/s/85295/images/icons/x-update-12x12.png \"Text created or updated\")\n* [adarshsolanki2004](https://codeforces.com/profile/adarshsolanki2004 \"Newbie adarshsolanki2004\") → [Cloudflare anti-bot verification not appearing](https://codeforces.com/blog/entry/151688) ![New comment\\(s\\)](//codeforces.com/codeforces.org/s/85295/images/icons/comment-12x12.png \"New comment(s)\")\n* [123gjweq2](https://codeforces.com/profile/123gjweq2 \"Master 123gjweq2\") → [Is this a good problem + can you please help me gauge its difficulty?](https://codeforces.com/blog/entry/156760) ![Text created or updated](//codeforces.com/codeforces.org/s/85295/images/icons/x-update-12x12.png \"Text created or updated\")\n\n| | [Detailed →](https://codeforces.com/recent-actions) |\n| | --------------------------------------------------- |\n\n* [nik\\_exists](https://codeforces.com/profile/nik%5Fexists)\n* [Blog](https://codeforces.com/blog/nik%5Fexists)\n* [Teams](https://codeforces.com/teams/with/nik%5Fexists)\n* [Submissions](https://codeforces.com/submissions/nik%5Fexists)\n* [Groups](https://codeforces.com/groups/with/nik%5Fexists)\n* [Contests](https://codeforces.com/contests/with/nik%5Fexists)\n* [Problemsetting](https://codeforces.com/contests/writer/nik%5Fexists)\n\n### [nik\\_exists's blog](https://codeforces.com/blog/nik%5Fexists)\n\n[ Codeforces Round 1119 (Div. 3) Editorial ](https://codeforces.com/blog/entry/156457) \n\nBy [nik\\_exists](https://codeforces.com/profile/nik%5Fexists \"Expert nik_exists\"), 12 days ago, ![In English](//codeforces.com/codeforces.org/s/85295/images/flags/24/gb.png \"In English\") \n\nThank y'all so much for participating!\n\n[2259A - Moo Language School](https://codeforces.com/contest/2259/problem/A \"Codeforces Round 1119 (Div. 3)\")\n\n**Solution**\n\nTutorial is loading...\n\n**Code**\n\n```\n#include \nusing namespace std;\n\nint main() {\n int t;\n cin >> t;\n\n while (t--) {\n int n, k;\n cin >> n >> k;\n\n string s;\n cin >> s;\n\n int ans = 0;\n for (int i = 0; i < n / k; i++) {\n bool has_non_nhoj_farm = false;\n for (int j = i * k; j < (i + 1) * k; j++) {\n if (s[j] == '0') has_non_nhoj_farm = true;\n }\n ans += !has_non_nhoj_farm;\n }\n\n cout << ans << \"\\n\";\n }\n}\n\n```\n\n**Difficulty**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too hard\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") A bit on the harder side\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good difficulty\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Easy\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too easy\n\n**Quality**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Amazing\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") So-so\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Bad\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Awful\n\n[2259B - Minus Two](https://codeforces.com/contest/2259/problem/B \"Codeforces Round 1119 (Div. 3)\")\n\n**Hint 1**\n\nWhen comparing two numbers that are initially not equal, the only way they can become equal after some number of operations is if at least one of them has an operation performed on it when it is $$$ \\\\lt 2$$$.\n\n**Hint 2**\n\nIf two elements are equal at some point, they'll be equal no matter how many more operations are performed. Therefore, it suffices to look at the end behavior of each element.\n\n**Solution**\n\nTutorial is loading...\n\n**Code**\n\n```\n#include \nusing namespace std;\n \nint main() {\n int t;\n cin >> t;\n \n while (t--) {\n int n;\n cin >> n;\n \n vector vec(n); for (auto &x: vec) cin >> x;\n \n int mod_4[] {0, 0, 0, 0};\n for (auto &x: vec) mod_4[x%4]++;\n cout << max({mod_4[0], mod_4[2], mod_4[1] + mod_4[3]}) << \"\\n\";\n }\n}\n\n```\n\n**Difficulty**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too hard\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") A bit on the harder side\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good difficulty\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Easy\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too easy\n\n**Quality**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Amazing\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") So-so\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Bad\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Awful\n\n[2259C - 101](https://codeforces.com/contest/2259/problem/C \"Codeforces Round 1119 (Div. 3)\")\n\n**Hint 1**\n\nIf a $$$-1$$$ is in between two $$$1$$$s, it would only hurt the answer to make it a $$$1$$$.\n\n**Hint 2**\n\nIf a $$$-1$$$ is not in between two $$$1$$$ s, the $$$-1$$$ would only be included in a subarray satisfying the conditions if it was an endpoint.\n\n**Solution**\n\nTutorial is loading...\n\n**Code**\n\n```\n#include \nusing namespace std;\n \nint main() {\n int t;\n cin >> t;\n \n while (t--) {\n int n;\n cin >> n;\n \n vector vec(n); for (auto &x: vec) cin >> x;\n \n for (int i = 0; i < n; i++) {\n if (vec[i] == -1) vec[i] = 1;\n if (vec[i] == 1) break;\n }\n \n for (int i = n-1; i >= 0; i--) {\n if (vec[i] == -1) vec[i] = 1;\n if (vec[i] == 1) break;\n }\n \n \n for (auto &x: vec) cout << max(x, 0) << \" \";\n cout << \"\\n\";\n }\n}\n\n```\n\n**Difficulty**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too hard\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") A bit on the harder side\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good difficulty\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Easy\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too easy\n\n**Quality**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Amazing\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") So-so\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Bad\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Awful\n\n[2259D - MEX Multiset](https://codeforces.com/contest/2259/problem/D \"Codeforces Round 1119 (Div. 3)\")\n\n**Hint 1**\n\nIf there is only one $$$0$$$ in the array, then exactly one set will have a mex greater than zero\n\n**Hint 2**\n\n$$$MEX(A) = MEX(B), MEX(C) = 0$$$ satisfies the constraints.\n\n**Solution**\n\nTutorial is loading...\n\n**Code**\n\n```\n#include \nusing namespace std;\n \nint main() {\n int t;\n cin >> t;\n \n while (t--) {\n int n;\n cin >> n;\n \n vector vec(n); for (auto &x: vec) cin >> x;\n if (count(vec.begin(), vec.end(), 0) == 1) {\n cout << \"NO\\n\";\n continue;\n }\n cout << \"YES\\n\";\n bool seen_zero = false;\n for (int i = 0; i < n; i++) {\n if (vec[i] != 0) cout << 'A';\n else if (seen_zero) cout << 'B';\n else {\n seen_zero = true;\n cout << 'C';\n }\n }\n cout << \"\\n\";\n }\n}\n\n```\n\n**Difficulty**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too hard\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") A bit on the harder side\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good difficulty\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Easy\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too easy\n\n**Quality**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Amazing\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") So-so\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Bad\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Awful\n\n[2259E - Treasure Map Destruction (Constructive Version)](https://codeforces.com/contest/2259/problem/E \"Codeforces Round 1119 (Div. 3)\")\n\n**Hint 1**\n\nIf an element $$$x$$$ at index $$$i$$$ is not equal to $$$-1$$$, there must be a treasure at either island $$$i + x$$$ or $$$i - x$$$\n\n**Hint 2**\n\nIf an element $$$x$$$ at index $$$i$$$ is not equal to $$$-1$$$, there must no treasures in the exclusive range of indices ($$$i + x$$$, $$$i - x$$$)\n\n**Solution**\n\nTutorial is loading...\n\n**Code**\n\n```\n#include \nusing namespace std;\n#define ll long long\n \nint main() {\n int t;\n cin >> t;\n \n \n while (t--) {\n int n;\n cin >> n;\n \n vector vec(n); for (auto &x: vec) cin >> x;\n \n vector diff(n+1);\n for (int i = 0; i < n; i++) {\n if (vec[i] > 0) {\n diff[max(0, i-vec[i]+1)]++; \n diff[min(n, i+vec[i])]--; \n }\n }\n vector restricted(n);\n int curr_count = 0;\n for (int i = 0; i < n; i++) {\n curr_count += diff[i];\n restricted[i] = curr_count > 0;\n }\n bool possible = true;\n for (int i = 0; i < n && possible; i++) {\n if (vec[i] >= 0) {\n if (i - vec[i] >= 0 && !restricted[i - vec[i]]) continue;\n if (i + vec[i] < n && !restricted[i + vec[i]]) continue;\n possible = false;\n }\n }\n \n if (!possible) cout << -1 << \"\\n\";\n else {\n for (auto x: restricted) cout << !x;\n cout << \"\\n\";\n } \n }\n}\n\n```\n\n**Difficulty**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too hard\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") A bit on the harder side\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good difficulty\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Easy\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too easy\n\n**Quality**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Amazing\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") So-so\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Bad\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Awful\n\n[2259F - Binary Bubble Sort Inversions](https://codeforces.com/contest/2259/problem/F \"Codeforces Round 1119 (Div. 3)\")\n\n**Hint 1**\n\nLook at performing some bubbles and reverse bubbles on small arrays. How many elements are moved?\n\n**Hint 2**\n\nA bubble will result is the leftmost 1 being moved all the way to the right. Similarly, a reverse bubble will result in the rightmost 0 being moved all the way to the left.\n\n**Hint 3**\n\nSince prefix zeroes and suffix ones don't matter, if we continuously remove them, then a bubble removes the last element of the array and a reverse bubble removes the first element of the array. Is there a way to remove the first and last elements of an array in constant time?\n\n**Solution**\n\nTutorial is loading...\n\n**Code**\n\n```\n#include \nusing namespace std;\n \n#define ll long long\nint main() {\n int t;\n cin >> t;\n \n while (t--) {\n int n;\n cin >> n;\n \n vector vec(n); for (auto &x: vec) cin >> x;\n string s;\n cin >> s;\n \n ll inversions = 0;\n int cnt_ones = 0;\n for (int i = 0; i < n; i++) {\n cnt_ones += vec[i];\n if (vec[i] == 0) inversions += cnt_ones;\n }\n cout << inversions << \" \";\n \n deque dq;\n for (auto &x: vec) dq.push_back(x);\n int cnt_zeroes = count(vec.begin(), vec.end(), 0);\n for (int i = 0; i < n; i++) {\n while (dq.size() > 0 && dq.front() == 0) {\n dq.pop_front();\n cnt_zeroes--;\n }\n while (dq.size() > 0 && dq.back() == 1) dq.pop_back();\n if (dq.size() == 0) {\n cout << 0 << \" \";\n continue;\n }\n if (s[i] == '0') {\n inversions -= (dq.size() - cnt_zeroes);\n dq.pop_back();\n cnt_zeroes--;\n }\n else {\n inversions -= cnt_zeroes;\n dq.pop_front();\n }\n cout << inversions << \" \";\n }\n cout << \"\\n\";\n }\n}\n\n```\n\n**Difficulty**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too hard\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") A bit on the harder side\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good difficulty\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Easy\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too easy\n\n**Quality**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Amazing\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") So-so\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Bad\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Awful\n\n[2259G - Index Removal](https://codeforces.com/contest/2259/problem/G \"Codeforces Round 1119 (Div. 3)\") \n\n**Hint 1**\n\nIgnoring indexes 1 and $$$n$$$, if we remove index $$$i$$$, if we perform an operation on index $$$j$$$, all elements between $$$\\[i, j\\]$$$ will also have an operation performed on them.\n\n**Hint 2**\n\nIf we remove index $$$i$$$, and we have to reduce the element at index $$$i+1$$$, $$$i+2$$$, etc, what will we subtract it to?\n\n**Solution**\n\nTutorial is loading...\n\n**Code**\n\n```\n#include \nusing namespace std;\n \n#define ll long long\nint main() {\n int t;\n cin >> t;\n \n while (t--) {\n ll n, k;\n cin >> n >> k;\n \n vector vec(n); for (auto &x: vec) cin >> x;\n vector vec_2(n); for (int i = 0; i < n; i++) vec_2[i] = vec[i] - i * k;\n \n vector prefix(n); prefix[0] = vec[0];\n for (int i = 1; i < n; i++) prefix[i] = prefix[i-1] + vec[i];\n \n vector ans(n);\n for (int i = 1; i < n-1; i++) {\n int lo = i+1;\n int hi = n-1;\n while (lo <= hi) {\n int mid = lo + (hi - lo) / 2;\n if (vec_2[mid] <= (vec[i-1] - i*k)) hi = mid - 1;\n else lo = mid + 1;\n }\n \n ll num_needed = (lo - i - 1);\n ans[i] = (prefix[num_needed + i] - prefix[i]) - (k * (num_needed * (num_needed + 1))/2) - vec[i-1] * num_needed;\n }\n for (auto &x: ans) cout << x << \" \";\n cout << \"\\n\";\n }\n}\n\n```\n\n**Difficulty**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too hard\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") A bit on the harder side\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good difficulty\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Easy\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too easy\n\n**Quality**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Amazing\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") So-so\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Bad\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Awful\n\n[2259H - Treasure Map Destruction (Counting Version)](https://codeforces.com/contest/2259/problem/H \"Codeforces Round 1119 (Div. 3)\")\n\nRead the solution to problem E first\n\n**Hint 1**\n\nLet's consider all locations that we have not restricted: we can separate them into 3 categories: the position is forced to be a treasure, the position has no undestroyed that depend on it, and all other positions. \n\n**Hint 2**\n\nThe islands that are considered \"non-restricted\" form a chain of locations, with each non-restricted chain having a treasure between them. What must be true of the locations in this chain such that no condition is violated?\n\n**Hint 3**\n\nFor every pair of adjacent locations, there must be at least one treasure at one of those locations.\n\n**Solution**\n\nTutorial is loading...\n\n**Code**\n\n```\n#include \nusing namespace std;\n#define ll long long\n \nconst ll MOD = 1000000007;\nconst int MAX_N = 200055;\nint main() {\n int t;\n cin >> t;\n \n \n vector fibo(MAX_N);\n fibo[1] = 1;\n fibo[2] = 1;\n for (int i = 3; i < MAX_N; i++) {\n fibo[i] = (fibo[i-1] + fibo[i-2]) % MOD;\n }\n \n while (t--) {\n int n;\n cin >> n;\n \n vector vec(n); for (auto &x: vec) cin >> x;\n \n vector diff(n+1);\n for (int i = 0; i < n; i++) {\n if (vec[i] > 0) {\n diff[max(0, i-vec[i]+1)]++; \n diff[min(n, i+vec[i])]--; \n }\n }\n \n vector restricted(n);\n vector forced(n);\n vector non_required;\n int curr_count = 0;\n for (int i = 0; i < n; i++) {\n curr_count += diff[i];\n restricted[i] = curr_count > 0;\n }\n \n bool possible = true;\n for (int i = 0; i < n; i++) {\n if (vec[i] != -1) {\n int cnt = 0;\n cnt += (i - vec[i] >= 0 && !restricted[i - vec[i]]); \n cnt += (i + vec[i] < n && !restricted[i + vec[i]]);\n if (cnt == 0) { possible = false;}\n if (cnt == 1 || vec[i] == 0) {\n if (i + vec[i] < n) forced[i + vec[i]] = true;\n if (i - vec[i] >= 0) forced[i - vec[i]] = true;\n }\n }\n }\n for (int i = 0; i < n; i++) {\n if (!restricted[i] && !forced[i]) non_required.push_back(i);\n }\n if (!possible) {\n cout << 0 << \"\\n\";\n continue;\n }\n \n ll total = 1;\n int chain_len = 1;\n for (int i = 0; i < non_required.size(); i++) {\n bool cond1 = i == non_required.size() - 1 || vec[(non_required[i] + non_required[i+1])/2] == (non_required[i+1] - non_required[i])/2;\n bool cond2 = i == non_required.size() - 1 || (non_required[i+1] - non_required[i]) % 2 == 0;\n //if (i != 1) cout << (non_required[i+1] - non_required[i]) << \"\\n\";\n if (i == non_required.size() - 1 || !cond1 || !cond2) {\n //cout << chain_len << \"\\n\";\n total *= fibo[chain_len+2];\n total %= MOD;\n chain_len = 1;\n }\n else chain_len++;\n }\n if (count(vec.begin(), vec.end(), -1) == n) {\n total = (total + MOD - 1) % MOD;\n }\n cout << total << \"\\n\";\n }\n}\n\n```\n\n**Difficulty**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too hard\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") A bit on the harder side\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good difficulty\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Easy\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too easy\n\n**Quality**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Amazing\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") So-so\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Bad\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Awful\n\n![](//codeforces.com/codeforces.org/s/85295/images/icons/paperclip-16x16.png) Tutorial of [Codeforces Round 1119 (Div. 3)](https://codeforces.com/contest/2259) \n\n* [![Vote: I like it](//codeforces.com/codeforces.org/s/85295/images/actions/voteup-gray.png \"Vote: I like it\")](#)\n* +202\n* [![Vote: I do not like it](//codeforces.com/codeforces.org/s/85295/images/actions/votedown-gray.png \"Vote: I do not like it\")](#)\n\n* [![Author](//codeforces.com/codeforces.org/s/85295/images/blog/user_16x16.png \"Author\")](https://codeforces.com/profile/nik%5Fexists) [nik\\_exists ](https://codeforces.com/profile/nik%5Fexists)\n* ![Publication date](//codeforces.com/codeforces.org/s/85295/images/blog/date_16x16.png \"Publication date\") 12 days ago\n* [![Comments](//codeforces.com/codeforces.org/s/85295/images/blog/comments_16x16.png \"Comments\")](https://codeforces.com/blog/entry/156457#comments) [118 ](https://codeforces.com/blog/entry/156457#comments)\n\n \n![Comments](//codeforces.com/codeforces.org/s/85295/images/icons/comments-48x48.png \"Comments\") Comments (118) \n\n[Write comment?](#) \n\n| » [nik\\_exists](https://codeforces.com/profile/nik%5Fexists \"Expert nik_exists\") | 12 days ago, [show (+1)](#) # |\n| -------------------------------------------------------------------------------- | ----------------------------- |\n\n| » [ ![](https://codeforces.com/userpic.codeforces.org/no-avatar.jpg) ](https://codeforces.com/profile/nik%5Fexists) [nik\\_exists](https://codeforces.com/profile/nik%5Fexists \"Expert nik_exists\") | 12 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156457?#comment-1390585 \"Link to comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/85295/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) +8 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/85295/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) _Auto comment: topic has been updated by [nik\\_exists](https://codeforces.com/profile/nik%5Fexists \"Expert nikexists\") ([previous revision](https://codeforces.com/topic/157099/en2), [new revision](https://codeforces.com/topic/157099/en3), [compare](https://codeforces.com/topic/157099/diff/en2/en3))._ [→](#) [Reply](#) |\n| -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n* | » » [wbw121124](https://codeforces.com/profile/wbw121124 \"Specialist wbw121124\") | 11 days ago, [show (+1)](#) # |\n| -------------------------------------------------------------------------------- | ----------------------------- |\n\n| » » [ ![](https://codeforces.com/userpic.codeforces.org/3260004/avatar/bfc5f8da6bb8c097.jpg) ](https://codeforces.com/profile/wbw121124) [wbw121124](https://codeforces.com/profile/wbw121124 \"Specialist wbw121124\") | 11 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156457?#comment-1390753 \"Link to comment\") [^](#comment-1390585 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/85295/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) +3 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/85295/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) where are codesit only have cpp [→](#) [Reply](#) |\n| --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n * | » » » [knkttp](https://codeforces.com/profile/knkttp \"Specialist knkttp\") | 11 days ago, [show](#) # |\n| ------------------------------------------------------------------------- | ------------------------ |\n\n| » » » [ ![](https://codeforces.com/userpic.codeforces.org/5193194/avatar/1319107b0a053e8e.jpg) ](https://codeforces.com/profile/knkttp) [knkttp](https://codeforces.com/profile/knkttp \"Specialist knkttp\") | 11 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156457?#comment-1390760 \"Link to comment\") [^](#comment-1390753 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/85295/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/85295/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) Wait until the hack end [→](#) [Reply](#) |\n| ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n * | » » » [nik\\_exists](https://codeforces.com/profile/nik%5Fexists \"Expert nik_exists\") | 10 days ago, [show (+1)](#) # |\n| ------------------------------------------------------------------------------------ | ----------------------------- |\n\n| » » » [ ![](https://codeforces.com/userpic.codeforces.org/no-avatar.jpg) ](https://codeforces.com/profile/nik%5Fexists) [nik\\_exists](https://codeforces.com/profile/nik%5Fexists \"Expert nik_exists\") | 10 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156457?#comment-1390952 \"Link to comment\") [^](#comment-1390753 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/85295/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/85295/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) Code has been added! [→](#) [Reply](#) |\n| ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n * | » » » » [wbw121124](https://codeforces.com/profile/wbw121124 \"Specialist wbw121124\") | 10 days ago, [show](#) # |\n| ------------------------------------------------------------------------------------ | ------------------------ |\n\n| » » » » [ ![](https://codeforces.com/userpic.codeforces.org/3260004/avatar/bfc5f8da6bb8c097.jpg) ](https://codeforces.com/profile/wbw121124) [wbw121124](https://codeforces.com/profile/wbw121124 \"Specialist wbw121124\") | 10 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156457?#comment-1391001 \"Link to comment\") [^](#comment-1390952 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/85295/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/85295/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) OK [→](#) [Reply](#) |\n| ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n| » [furqansuleymanli](https://codeforces.com/profile/furqansuleymanli \"Newbie furqansuleymanli\") | 12 days ago, [show (+1)](#) # |\n| ----------------------------------------------------------------------------------------------- | ----------------------------- |\n\n| » [ ![](https://codeforces.com/userpic.codeforces.org/5563911/avatar/f4984ba07dcb9e51.jpg) ](https://codeforces.com/profile/furqansuleymanli) [furqansuleymanli](https://codeforces.com/profile/furqansuleymanli \"Newbie furqansuleymanli\") | 12 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156457?#comment-1390593 \"Link to comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/85295/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) +2 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/85295/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) good contest! [→](#) [Reply](#) |\n| ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n* | » » [nik\\_exists](https://codeforces.com/profile/nik%5Fexists \"Expert nik_exists\") | 12 days ago, [show](#) # |\n| ---------------------------------------------------------------------------------- | ------------------------ |\n\n| » » [ ![](https://codeforces.com/userpic.codeforces.org/no-avatar.jpg) ](https://codeforces.com/profile/nik%5Fexists) [nik\\_exists](https://codeforces.com/profile/nik%5Fexists \"Expert nik_exists\") | 12 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156457?#comment-1390642 \"Link to comment\") [^](#comment-1390593 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/85295/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) +7 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/85295/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) tysm! [→](#) [Reply](#) |\n| ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ |\n* | » » [dhiruuu\\_singh26](https://codeforces.com/profile/dhiruuu%5Fsingh26 \"Newbie dhiruuu_singh26\") | 11 days ago, [show](#) # |\n| ------------------------------------------------------------------------------------------------- | ------------------------ |\n\n| » » [ ![](https://codeforces.com/userpic.codeforces.org/4933574/avatar/bca47b478c630dee.jpg) ](https://codeforces.com/profile/dhiruuu%5Fsingh26) [dhiruuu\\_singh26](https://codeforces.com/profile/dhiruuu%5Fsingh26 \"Newbie dhiruuu_singh26\") | 11 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156457?#comment-1390883 \"Link to comment\") [^](#comment-1390593 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/85295/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/85295/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) TIGER! TIGER!........ [→](#) [Reply](#) |\n| ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n* | » » [While.1](https://codeforces.com/profile/While.1 \"Newbie While.1\") | 10 days ago, [show](#) # |\n| ---------------------------------------------------------------------- | ------------------------ |\n\n| » » [ ![](https://codeforces.com/userpic.codeforces.org/5072827/avatar/578ea0266e326443.jpg) ](https://codeforces.com/profile/While.1) [While.1](https://codeforces.com/profile/While.1 \"Newbie While.1\") | 10 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156457?#comment-1390954 \"Link to comment\") [^](#comment-1390593 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/85295/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/85295/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) felt that problem F was more trickier than problem G! [→](#) [Reply](#) |\n| --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n| » [why127pan](https://codeforces.com/profile/why127pan \"Newbie why127pan\") | 12 days ago, [show (+1)](#) # |\n| -------------------------------------------------------------------------- | ----------------------------- |\n\n| » [ ![](https://codeforces.com/userpic.codeforces.org/5327650/avatar/5df2e47dfbff541b.jpg) ](https://codeforces.com/profile/why127pan) [why127pan](https://codeforces.com/profile/why127pan \"Newbie why127pan\") | 12 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156457?#comment-1390594 \"Link to comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/85295/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/85295/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) nice round thanks ! [→](#) [Reply](#) |\n| --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n* | » » [nik\\_exists](https://codeforces.com/profile/nik%5Fexists \"Expert nik_exists\") | 12 days ago, [show](#) # |\n| ---------------------------------------------------------------------------------- | ------------------------ |\n\n| » » [ ![](https://codeforces.com/userpic.codeforces.org/no-avatar.jpg) ](https://codeforces.com/profile/nik%5Fexists) [nik\\_exists](https://codeforces.com/profile/nik%5Fexists \"Expert nik_exists\") | 12 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156457?#comment-1390645 \"Link to comment\") [^](#comment-1390594 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/85295/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) +6 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/85295/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) tysm! [→](#) [Reply](#) |\n| ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ |\n\n| » [megahertz13](https://codeforces.com/profile/megahertz13 \"Pupil megahertz13\") | 12 days ago, [show (+1)](#) # |\n| ------------------------------------------------------------------------------- | ----------------------------- |\n\n| » [ ![](https://codeforces.com/userpic.codeforces.org/2827759/avatar/2d153e90dd0538bd.jpg) ](https://codeforces.com/profile/megahertz13) [megahertz13](https://codeforces.com/profile/megahertz13 \"Pupil megahertz13\") | 12 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156457?#comment-1390596 \"Link to comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/85295/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/85295/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) good problems, C was kind of annoying to implement [→](#) [Reply](#) |\n\n...201282 bytes truncated...\n\nThe tool call succeeded but the output was truncated. Full output saved to: /Users/zayd/.local/share/opencode/tool-output/tool_0adf2967d001uMo7LeOpRkFMSn\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_0adf2967d001uMo7LeOpRkFMSn" }, "title": "https://codeforces.com/blog/entry/156457 (text/markdown; charset=utf-8)", "time": { "start": 1789624746046, "end": 1789624751744 } }, "metadata": { "openai": { "itemId": "fc_011d94407418b0c5016aab81a92cc887d1bfac8802d3e38497" } }, "id": "prt_0adf27d05001FlbQ4lJOQ04ncM", "sessionID": "ses_f520d9080ffeOzR9O7LuPqvgeA", "messageID": "msg_0adf270120010LiuZDmVLbToj2" }, { "reason": "tool-calls", "type": "step-finish", "tokens": { "total": 9152, "input": 9120, "output": 32, "reasoning": 0, "cache": { "write": 0, "read": 0 } }, "cost": 0, "id": "prt_0adf29683001NIpcWaOq1zNq3H", "sessionID": "ses_f520d9080ffeOzR9O7LuPqvgeA", "messageID": "msg_0adf270120010LiuZDmVLbToj2" } ] }, { "info": { "parentID": "msg_0adf26f91001j25fyMcm8mcDRY", "role": "assistant", "mode": "nudge-generation", "agent": "nudge-generation", "variant": "low", "path": { "cwd": "/private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-QYeIIC", "root": "/" }, "cost": 0, "tokens": { "total": 23417, "input": 13221, "output": 1339, "reasoning": 25, "cache": { "write": 0, "read": 8832 } }, "modelID": "gpt-6-astra", "providerID": "openai", "time": { "created": 1789624751752, "completed": 1789624797284 }, "finish": "tool-calls", "structured": { "status": "success", "reason": null, "hints": [ { "order": 1, "content": "You do **not** need to maximize any MEX. Small MEX values are perfectly fine. What makes a multiset's MEX equal to $0$ or $1$?" }, { "order": 2, "content": "Focus on the number of zeros. If there are no zeros, every multiset has MEX $0$. If there is exactly one zero, how many multisets can have positive MEX?" }, { "order": 3, "content": "With exactly one zero, the MEX values must be $m,0,0$ in some order, where $m\\ge 1$. The required inequality becomes $m\\ge 2m$, which is impossible." }, { "order": 4, "content": "For at least two zeros, aim for MEX values $1,1,0$. They satisfy the inequality with equality. How can you keep the first two multisets from containing $1$, while keeping the third from containing $0$?" }, { "order": 5, "content": "Put the first zero into $A$, every remaining zero into $B$, and every nonzero element into $C$. With at least two zeros, the MEX values are exactly $1,1,0$. With no zeros, the same rule gives $0,0,0$. Therefore, output `NO` exactly when there is one zero; otherwise, output this construction." } ], "editorial": "The trick is to keep the MEX values tiny. We aren't trying to maximize anything—large MEX values just create extra work.\n\n## Key insight\n\nTwo facts are enough:\n\n- A multiset without $0$ has MEX $0$, even if it contains lots of other numbers.\n- A nonempty multiset containing only zeros has MEX $1$.\n\nLet $z$ be the number of zeros in the array. Everything comes down to three cases.\n\n### No zeros: always possible\n\nNo multiset can contain $0$, so all three MEX values are $0$. The condition becomes\n\n$$0\\ge 0.$$\n\nWe can put every element into $C$. Empty multisets are allowed.\n\n### Exactly one zero: impossible\n\nThat zero must go into exactly one multiset. Its MEX is some $m\\ge 1$, while the other two multisets have MEX $0$.\n\nThe required inequality would be\n\n$$m+0+0\\ge 2m,$$\n\nwhich is false for every positive $m$. No clever placement of the other numbers can fix this.\n\n### At least two zeros: construct MEX values $1,1,0$\n\nUse this distribution:\n\n- Put the first zero into $A$.\n- Put every remaining zero into $B$.\n- Put every nonzero element into $C$.\n\nBoth $A$ and $B$ are nonempty and contain only zeros, so their MEX values are $1$. Multiset $C$ contains no zero, so its MEX is $0$.\n\nThus,\n\n$$1+1+0=2=2\\cdot\\max(1,1,0).$$\n\nThe nonzero values can be anything. Once $C$ is missing zero, they have no effect on its MEX.\n\n## Algorithm\n\nBuild the answer string while reading the array:\n\n1. Initialize every character to `C` and set a zero counter to $0$.\n2. For each zero, increment the counter. Assign the first zero to `A` and all later zeros to `B`.\n3. After reading all elements:\n - If the zero counter is exactly $1$, print `NO`.\n - Otherwise, print `YES` and the answer string.\n\nThere is no need to store the array, sort it, or calculate any MEX explicitly.\n\n## Correctness proof\n\nWe prove that the algorithm outputs `NO` exactly when no valid distribution exists, and otherwise outputs a valid distribution.\n\nIf there is exactly one zero, exactly one multiset has positive MEX $m$, and the other two have MEX $0$. Since $m\\ge 1$, the inequality $m\\ge 2m$ cannot hold. Therefore, the algorithm correctly outputs `NO`.\n\nIf there are no zeros, the algorithm puts all elements into $C$. All three multisets have MEX $0$, so the required inequality holds.\n\nIf there are at least two zeros, the algorithm gives $A$ the first zero, $B$ all remaining zeros, and $C$ all nonzero elements. Their MEX values are respectively $1,1,0$, which satisfy the required inequality with equality.\n\nIn both successful cases, every array element is assigned to exactly one multiset. These cases cover every possible zero count, so the algorithm is correct.\n\n## Complexity\n\nEach test case takes $O(n)$ time and $O(n)$ space for the output string. Apart from that string, the algorithm uses $O(1)$ extra space.", "solution": "#include \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\n string s(n, 'C');\n int zeros = 0;\n for (int i = 0; i < n; ++i) {\n int x;\n cin >> x;\n if (x == 0) {\n ++zeros;\n s[i] = (zeros == 1 ? 'A' : 'B');\n }\n }\n\n if (zeros == 1) {\n cout << \"NO\\n\";\n } else {\n cout << \"YES\\n\" << s << '\\n';\n }\n }\n}\n" }, "id": "msg_0adf2968800159kxkzIZo2K8f2", "sessionID": "ses_f520d9080ffeOzR9O7LuPqvgeA" }, "parts": [ { "type": "step-start", "id": "prt_0adf2a7da001bN8C931VJu9UDF", "sessionID": "ses_f520d9080ffeOzR9O7LuPqvgeA", "messageID": "msg_0adf2968800159kxkzIZo2K8f2" }, { "type": "reasoning", "text": "**Finalizing five hints**", "time": { "start": 1789624756188, "end": 1789624758374 }, "metadata": { "openai": { "itemId": "rs_0e17ef975c21db9b016aab81b4244087d2a4f4e31ca185d6bc", "reasoningEncryptedContent": "gAAAAABqq4G2MRA4wb2bkQ5B-O4x1VogqrvDuPxgYm4gt6aJmyhs2-xPRP1ilanv1pgqm85INGTa5qmPK9GeRx2t7x2Rx5bfNW8h3r4owj8m6hc7xfpYvfb-jr9YRwW3AV6oTq_UR7XcyQIsc_Dtwu-_IKWejSPQHLoOwNC4sO5g_eapijY-LOVgkm2OY6R290Bink4nJjm6IWkAgdTQr9go7Yk8BinxIKH1cZNjxVqFT9vuf5FZK_P3gX0oN6Rmt1PeDqmMuHba-IFNxSneEjbmqE2alwfWLS_3eO3gyvar_tpIwfOQIju5dP7G2ofEmpKg4woqlopmqHycVbJL8HXN3rP6YDAQujx4BfHO8Da2H1qfUkMiDt4lD98iGvF8MbPICgFI2CJxDu3OzfX9dcmXEBkUoYEPW6Qmk9qgIgH8Vb91OskayK0jDC4eelBtNfc-hH7sXhBpUjTCKpTGHWONhH6l4n6Mi0WuiGisCAOfEZhNfc7kGukHSSLaOIG7xYhEJ9PrUmR8bsqOK-26BowuKgZeYuPz5CuVqTj9cnDnkZ8Gew3SXrQtl9HFjp7FzIPk-_GtQIzIhkL_gwLk4-kUaw1cHKZ84paJtuye1lNE-JWmwk3FmB1PL4LB89XmJCprN13wQSF2cuxuNYDvof8VWk2fPVkAoTO3xpeN-LFXk7dhZUj6IFJUp3Dabyfyk29FiCbW2PWig6DzHMkXltYSvtef8ISgJKD5aRHykEnOnqXi95vA6y7SwwCMD21zk02AGbieWGIs4LD5cc7gOJioxRa05hk0yVbaKbxXW1SJrzBA8BnfWIE9J6gs3UcNh2jtaX1M5rpb7dCmqbVUCzYA5sPnEf8JKsedV7vRLxd6GiG5XwjtCRDFFfH1g-nklbjyqhRlE60XBB8GoEPg8QMduJJE9bQJTKqniXxgsZzSGPvecitQJv1H6Ej5vhDAU9wL5MTxgrhvcyIBw-mJfxuo9YF6ZPulGxag3gYVnCZgrP-Oyj74Pscg19_fyt4-gwcE8OXgzgT5cUgYyL3GL70hY7MKESQM1-ehLFASSUv9FqVzu-hvn_twKA6FpsfnCnACNhEpxwsCEnXObngo8uN3p1FUmipdvtXgZrixV-dNPDkDY9NC7ai6Ta6tD758Rds1qNqBqPAmZ90WDpDzlw_pxqypgplAePymnCEevPmkQw5gHoS4lmnB7QrRScdLXtVgYvRpQ5xfWNfIRPtlK1pVEyivBXQYZGsV0MsHI4qkKJyOPxF4F6fQfqJfxslOCW3v_NyxNA2Lh0Bvd0XPQNqh6kxUv8KoxmV37Y2CmCZasWUe7hfhyCFWAgn4LS0NAd9TouEZ622p6YleXCvLhEcVSjl6rx_hK9yBTkk3POq1OaexAmkkxUZ4ofY6uz2t1UdyeksEsTRNtG2H0qdH6n-yKec9hkzbGut0vBOImqvUY_xlx3JGVi7mopN5bdOe-R3i_5P5z0J7" } }, "id": "prt_0adf2a7dc0014xwyUBl45X1T1u", "sessionID": "ses_f520d9080ffeOzR9O7LuPqvgeA", "messageID": "msg_0adf2968800159kxkzIZo2K8f2" }, { "type": "tool", "tool": "StructuredOutput", "callID": "call_cwZndcV8HI7bUMLXorCGXWWb", "state": { "status": "completed", "input": { "status": "success", "reason": null, "hints": [ { "order": 1, "content": "You do **not** need to maximize any MEX. Small MEX values are perfectly fine. What makes a multiset's MEX equal to $0$ or $1$?" }, { "order": 2, "content": "Focus on the number of zeros. If there are no zeros, every multiset has MEX $0$. If there is exactly one zero, how many multisets can have positive MEX?" }, { "order": 3, "content": "With exactly one zero, the MEX values must be $m,0,0$ in some order, where $m\\ge 1$. The required inequality becomes $m\\ge 2m$, which is impossible." }, { "order": 4, "content": "For at least two zeros, aim for MEX values $1,1,0$. They satisfy the inequality with equality. How can you keep the first two multisets from containing $1$, while keeping the third from containing $0$?" }, { "order": 5, "content": "Put the first zero into $A$, every remaining zero into $B$, and every nonzero element into $C$. With at least two zeros, the MEX values are exactly $1,1,0$. With no zeros, the same rule gives $0,0,0$. Therefore, output `NO` exactly when there is one zero; otherwise, output this construction." } ], "editorial": "The trick is to keep the MEX values tiny. We aren't trying to maximize anything—large MEX values just create extra work.\n\n## Key insight\n\nTwo facts are enough:\n\n- A multiset without $0$ has MEX $0$, even if it contains lots of other numbers.\n- A nonempty multiset containing only zeros has MEX $1$.\n\nLet $z$ be the number of zeros in the array. Everything comes down to three cases.\n\n### No zeros: always possible\n\nNo multiset can contain $0$, so all three MEX values are $0$. The condition becomes\n\n$$0\\ge 0.$$\n\nWe can put every element into $C$. Empty multisets are allowed.\n\n### Exactly one zero: impossible\n\nThat zero must go into exactly one multiset. Its MEX is some $m\\ge 1$, while the other two multisets have MEX $0$.\n\nThe required inequality would be\n\n$$m+0+0\\ge 2m,$$\n\nwhich is false for every positive $m$. No clever placement of the other numbers can fix this.\n\n### At least two zeros: construct MEX values $1,1,0$\n\nUse this distribution:\n\n- Put the first zero into $A$.\n- Put every remaining zero into $B$.\n- Put every nonzero element into $C$.\n\nBoth $A$ and $B$ are nonempty and contain only zeros, so their MEX values are $1$. Multiset $C$ contains no zero, so its MEX is $0$.\n\nThus,\n\n$$1+1+0=2=2\\cdot\\max(1,1,0).$$\n\nThe nonzero values can be anything. Once $C$ is missing zero, they have no effect on its MEX.\n\n## Algorithm\n\nBuild the answer string while reading the array:\n\n1. Initialize every character to `C` and set a zero counter to $0$.\n2. For each zero, increment the counter. Assign the first zero to `A` and all later zeros to `B`.\n3. After reading all elements:\n - If the zero counter is exactly $1$, print `NO`.\n - Otherwise, print `YES` and the answer string.\n\nThere is no need to store the array, sort it, or calculate any MEX explicitly.\n\n## Correctness proof\n\nWe prove that the algorithm outputs `NO` exactly when no valid distribution exists, and otherwise outputs a valid distribution.\n\nIf there is exactly one zero, exactly one multiset has positive MEX $m$, and the other two have MEX $0$. Since $m\\ge 1$, the inequality $m\\ge 2m$ cannot hold. Therefore, the algorithm correctly outputs `NO`.\n\nIf there are no zeros, the algorithm puts all elements into $C$. All three multisets have MEX $0$, so the required inequality holds.\n\nIf there are at least two zeros, the algorithm gives $A$ the first zero, $B$ all remaining zeros, and $C$ all nonzero elements. Their MEX values are respectively $1,1,0$, which satisfy the required inequality with equality.\n\nIn both successful cases, every array element is assigned to exactly one multiset. These cases cover every possible zero count, so the algorithm is correct.\n\n## Complexity\n\nEach test case takes $O(n)$ time and $O(n)$ space for the output string. Apart from that string, the algorithm uses $O(1)$ extra space.", "solution": "#include \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\n string s(n, 'C');\n int zeros = 0;\n for (int i = 0; i < n; ++i) {\n int x;\n cin >> x;\n if (x == 0) {\n ++zeros;\n s[i] = (zeros == 1 ? 'A' : 'B');\n }\n }\n\n if (zeros == 1) {\n cout << \"NO\\n\";\n } else {\n cout << \"YES\\n\" << s << '\\n';\n }\n }\n}\n" }, "output": "Structured output captured successfully.", "metadata": { "valid": true }, "title": "Structured Output", "time": { "start": 1789624797177, "end": 1789624797180 } }, "metadata": { "openai": { "itemId": "fc_0e17ef975c21db9b016aab81b6405887d2aa5ff7c5a70c734e" } }, "id": "prt_0adf2b067001xxwBmx4vXJvzUb", "sessionID": "ses_f520d9080ffeOzR9O7LuPqvgeA", "messageID": "msg_0adf2968800159kxkzIZo2K8f2" }, { "reason": "tool-calls", "type": "step-finish", "tokens": { "total": 23417, "input": 13221, "output": 1339, "reasoning": 25, "cache": { "write": 0, "read": 8832 } }, "cost": 0, "id": "prt_0adf34862001vBNhVcKJJtXtb0", "sessionID": "ses_f520d9080ffeOzR9O7LuPqvgeA", "messageID": "msg_0adf2968800159kxkzIZo2K8f2" } ] } ] }