{ "info": { "id": "ses_f520d4942ffeX6i2Z0GTu8Gr9y", "slug": "eager-island", "projectID": "global", "directory": "/private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-jbEoOV", "path": "private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-jbEoOV", "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": 22429, "output": 1650, "reasoning": 13, "cache": { "read": 29568, "write": 0 } }, "time": { "created": 1789624759997, "updated": 1789624830349 } }, "messages": [ { "info": { "role": "user", "time": { "created": 1789624760013 }, "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_0adf2b6cd001kvShaifNWjqoa5", "sessionID": "ses_f520d4942ffeX6i2Z0GTu8Gr9y" }, "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 2259B: \"Minus Two\" (rated 800).\n\nKnown problem metadata:\n- Contest ID: 2259\n- Index: B\n- Name: Minus Two\n- Rating: 800 (Codeforces difficulty rating)\n- Tags: math, number theory\n- Problem URL: https://codeforces.com/contest/2259/problem/B\n\nUse the rating and tags as weak signals only. The supplied statement is the source of truth.\n\nProblem Statement:\n\n
B. Minus Two
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

You are given an array $$$a_1, a_2, \\ldots, a_n$$$. You may perform the following operation:

Find the maximum possible frequency of any integer in $$$a$$$ after performing the operation an arbitrary number (possibly zero) of times.

Input

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

The first line of each test case contains an integer $$$n$$$ ($$$1 \\leq n \\leq 2 \\cdot 10^5$$$) — the length of $$$a$$$.

The second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \\ldots, a_n$$$ ($$$1 \\leq a_i \\leq 10^9$$$) — the array $$$a$$$.

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

Output

For each test case, output a single integer denoting the maximum possible frequency of any integer in $$$a$$$ after performing the operation an arbitrary number of times.

Example
Input
5
2
1 3
4
1 1 1 2
3
6 7 8
4
2 2 2 2
5
1 10 100 1000 100000
Output
2
3
1
4
3
Note

In the first test case, we can perform the operation once to have $$$a = [|1 - 2|, |3 - 2|] = [1, 1]$$$, which means the maximum possible frequency of any integer in $$$a$$$ is $$$2$$$.

In the second test case, the maximum frequency of an integer in $$$a$$$ is $$$3$$$. It can be shown that, no matter how many operations we perform, the maximum possible frequency of an integer in $$$a$$$ will never exceed $$$3$$$.

\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 2259B)\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_0adf2b6ce001jGvzNJwET2KyqP", "sessionID": "ses_f520d4942ffeX6i2Z0GTu8Gr9y", "messageID": "msg_0adf2b6cd001kvShaifNWjqoa5" } ] }, { "info": { "parentID": "msg_0adf2b6cd001kvShaifNWjqoa5", "role": "assistant", "mode": "nudge-generation", "agent": "nudge-generation", "variant": "low", "path": { "cwd": "/private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-jbEoOV", "root": "/" }, "cost": 0, "tokens": { "total": 8671, "input": 8639, "output": 32, "reasoning": 0, "cache": { "write": 0, "read": 0 } }, "modelID": "gpt-6-astra", "providerID": "openai", "time": { "created": 1789624760143, "completed": 1789624768731 }, "finish": "tool-calls", "id": "msg_0adf2b74f001Bdfphm9ftQjxJs", "sessionID": "ses_f520d4942ffeX6i2Z0GTu8Gr9y" }, "parts": [ { "type": "step-start", "id": "prt_0adf2c0fb0016SrRQkKDMVei5w", "sessionID": "ses_f520d4942ffeX6i2Z0GTu8Gr9y", "messageID": "msg_0adf2b74f001Bdfphm9ftQjxJs" }, { "type": "tool", "tool": "webfetch", "callID": "call_sGg82Wehz1L527Q8AOabRfuS", "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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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_0adf2d8d4001YOVnXrBu3QXF3t\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_0adf2d8d4001YOVnXrBu3QXF3t" }, "title": "https://codeforces.com/blog/entry/156457 (text/markdown; charset=utf-8)", "time": { "start": 1789624763417, "end": 1789624768726 } }, "metadata": { "openai": { "itemId": "fc_0ad1613d28f1bb88016aab81ba8df487d19dca35b929003284" } }, "id": "prt_0adf2c0fc001g88goO55x5zx81", "sessionID": "ses_f520d4942ffeX6i2Z0GTu8Gr9y", "messageID": "msg_0adf2b74f001Bdfphm9ftQjxJs" }, { "reason": "tool-calls", "type": "step-finish", "tokens": { "total": 8671, "input": 8639, "output": 32, "reasoning": 0, "cache": { "write": 0, "read": 0 } }, "cost": 0, "id": "prt_0adf2d8d80017E0bb6SwtMz6I3", "sessionID": "ses_f520d4942ffeX6i2Z0GTu8Gr9y", "messageID": "msg_0adf2b74f001Bdfphm9ftQjxJs" } ] }, { "info": { "parentID": "msg_0adf2b6cd001kvShaifNWjqoa5", "role": "assistant", "mode": "nudge-generation", "agent": "nudge-generation", "variant": "low", "path": { "cwd": "/private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-jbEoOV", "root": "/" }, "cost": 0, "tokens": { "total": 21762, "input": 13255, "output": 174, "reasoning": 13, "cache": { "write": 0, "read": 8320 } }, "modelID": "gpt-6-astra", "providerID": "openai", "time": { "created": 1789624768732, "completed": 1789624776525 }, "finish": "tool-calls", "id": "msg_0adf2d8dc001XoGqgLiEvU2jPS", "sessionID": "ses_f520d4942ffeX6i2Z0GTu8Gr9y" }, "parts": [ { "type": "step-start", "id": "prt_0adf2e303001D4UZDZTEmsFrCF", "sessionID": "ses_f520d4942ffeX6i2Z0GTu8Gr9y", "messageID": "msg_0adf2d8dc001XoGqgLiEvU2jPS" }, { "type": "reasoning", "text": "**Verifying brute-force approach**", "time": { "start": 1789624771333, "end": 1789624773658 }, "metadata": { "openai": { "itemId": "rs_0a3f5db65a6911aa016aab81c2a2c887d28cfa67978f29b686", "reasoningEncryptedContent": "gAAAAABqq4HFz8yqJiSz3wmbAuE5Pm7vHU5J8m8LqFluX57QquIg2a19rSooX4FFEOb13Iq0E9aa-ATXUqMJ_6aQI8cbWr_JXtdD6Onx7j5tt9WZwTTBOEF8LtbZFCdYeDbFtY74fZdTxqELQyZ2t3lkl6gEdvFwoWWm5hWtTEcFWQ8a-vgCjRQBz_LmSCNEujIZSBWyrW-Yy8Z0Mm5OaVFNYrZ3IIcrgEs6Q0qU9arX03uSqjtsCEHiQKMP4IDO7_ofDGEzJRTUYpG-7gg5tucE1JEcJbB8hFuB6YcWpn_z59Hd5Fe_AZa5LMN6aGEKvYMEJ-0-kVJXrGQ7gEtHdp83aPcOckgtD84lTzibhlBv4tUhdYmSZNfq6Yj3Mmhu9UmbxTu0pGWmIJRLHj-T1wx7Wbwh2P-CMwPeSRg5IugaWhoohe-pNtIYkS74lSO90Eyr9iSD-KUEfJ4vkgS7eei18knktz280VDxpKjpuL2mx6gnH_cZ5nOClqs1LUWCiC7S8Jd-I998xrmSdR3zxwU2nH3cV0sYYCb8A87YY6e9s3FifG7S4giQn-JmRdti8gqv9J8ZKV4V1Qt8CpG1E7AaXzbSCpgYTKtOU1Jm4kqFNRilW66B4M8yKiKCLoa0ByhnzqgAqmbnpimR0GFxpOdiB5HM9o0emHQmxK2RLB1D5gu5szJw4ChT67JhaJaOytz1jDgxVKPpdpxHRS3zrpzizaV-xKdDeF4_tnZwTjnypoKu6CpW_G5jw5ul-QDrzLJPFIz_jfcOU7KIPENdi_zIxbt1bw05opDgN-rDmwllhxM5-tKELSg9R2feSw8mMMCSOjK5F40BbZU44AfVgxHPJvqpicX6wPAvS0lmdL9wKS88xRTENy7coeD5euNvOZPwf0bx8wf9hATw3EWtv8etCHBt2e7_hryP1_ZTz_BzYDN7BrZzr98X9L54MBoNmWscv7nsK9Ss3rUMe4yL_nlE5dBtXqZS5IivHcmYTn59kVrleFC4CCMopc9ps3YGXWDZPnR5s5UmXpZh_tUfKuWjue1MTJfZElTdD8R1mfHP7g6MMMKJNiOlpYj1wmdCoEzroLnciEEtJvEdj1MQEK_SeKvJhxZ0UhLevEXnay0g5hAhkCi1lFxT5WouLgFbn5H4a6NlTN0rDKDeZmEIH45cMZGuaDeEcsXcyVsI89oWSSPfwavW-_wc6Rwmepztmu_RgnfaGbldxsEDJb0WsjOKqUW90qgKrmnzc63Vlrzfc4A7-_nmC7cti_1JzpVFWWRPXhKBc_zpouesC_C5jUoPFzGXbo81V1wd7dEBK0cVW7vUzrJm3FQwLafF0-X6zey6Ix9YENrt" } }, "id": "prt_0adf2e305001bfOrgFh2Y5jD0h", "sessionID": "ses_f520d4942ffeX6i2Z0GTu8Gr9y", "messageID": "msg_0adf2d8dc001XoGqgLiEvU2jPS" }, { "type": "tool", "tool": "bash", "callID": "call_knhOvHoWxkmis9dhkE9s3lZb", "state": { "status": "completed", "input": { "command": "python3 -c 'from itertools import combinations_with_replacement; from collections import Counter\nchecked=0\nfor n in range(1,6):\n for a in combinations_with_replacement(range(1,13),n):\n c=Counter(x%4 for x in a); expected=max(c[0],c[2],c[1]+c[3]); b=list(a); best=0\n for _ in range(9):\n best=max(best,max(Counter(b).values())); b=[abs(x-2) for x in b]\n assert best==expected,(a,best,expected)\n checked+=1\nprint(f\"Verified {checked} small arrays against direct simulation.\")'" }, "output": "Verified 6187 small arrays against direct simulation.\n", "metadata": { "output": "Verified 6187 small arrays against direct simulation.\n", "exit": 0, "truncated": false }, "title": "python3 -c 'from itertools import combinations_with_replacement; from collections import Counter\nchecked=0\nfor n in range(1,6):\n for a in combinations_with_replacement(range(1,13),n):\n c=Counter(x%4 for x in a); expected=max(c[0],c[2],c[1]+c[3]); b=list(a); best=0\n for _ in range(9):\n best=max(best,max(Counter(b).values())); b=[abs(x-2) for x in b]\n assert best==expected,(a,best,expected)\n checked+=1\nprint(f\"Verified {checked} small arrays against direct simulation.\")'", "time": { "start": 1789624776435, "end": 1789624776523 } }, "metadata": { "openai": { "itemId": "fc_0a3f5db65a6911aa016aab81c57e6487d2abad9ef82f2665c2" } }, "id": "prt_0adf2ec1b001cr5Mx1h7ODenkA", "sessionID": "ses_f520d4942ffeX6i2Z0GTu8Gr9y", "messageID": "msg_0adf2d8dc001XoGqgLiEvU2jPS" }, { "reason": "tool-calls", "type": "step-finish", "tokens": { "total": 21762, "input": 13255, "output": 174, "reasoning": 13, "cache": { "write": 0, "read": 8320 } }, "cost": 0, "id": "prt_0adf2f74c001aNzd5F0YRcpZns", "sessionID": "ses_f520d4942ffeX6i2Z0GTu8Gr9y", "messageID": "msg_0adf2d8dc001XoGqgLiEvU2jPS" } ] }, { "info": { "parentID": "msg_0adf2b6cd001kvShaifNWjqoa5", "role": "assistant", "mode": "nudge-generation", "agent": "nudge-generation", "variant": "low", "path": { "cwd": "/private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-jbEoOV", "root": "/" }, "cost": 0, "tokens": { "total": 23227, "input": 535, "output": 1444, "reasoning": 0, "cache": { "write": 0, "read": 21248 } }, "modelID": "gpt-6-astra", "providerID": "openai", "time": { "created": 1789624776526, "completed": 1789624830346 }, "finish": "tool-calls", "structured": { "status": "success", "reason": null, "hints": [ { "order": 1, "content": "Every operation changes **all** elements together. Try following a few small values until their behavior repeats. What happens to $1$, $2$, $3$, and $4$?" }, { "order": 2, "content": "Every odd value eventually reaches $1$ and stays there. Every even value eventually enters the cycle $0\\to2\\to0$. But do all even values reach the same point in that cycle at the same time?" }, { "order": 3, "content": "Compare starting values $2$ and $4$: they never become equal. Now compare $2$ and $6$: they eventually do. For an even starting value $x$, reaching $0$ takes $x/2$ operations. The parity of that arrival time matters." }, { "order": 4, "content": "Split the original elements into three groups: odd values, values divisible by $4$, and values congruent to $2$ modulo $4$. After enough operations, each group consists of equal values, and the three groups have different values." }, { "order": 5, "content": "Once two elements become equal, they stay equal forever because they always receive the same operation. So no earlier moment can beat the maximum frequency after everything reaches its final fixed point or cycle. If $c_r$ counts original values with remainder $r$ modulo $4$, the answer is $\\max(c_1+c_3,c_0,c_2)$." } ], "editorial": "The catch is **“for all indices.”** You cannot stop one element while another catches up. In particular, “all even numbers eventually become equal” is false: $2$ and $4$ keep missing each other forever.\n\n### What happens to one value?\n\nWhile $x\\ge2$, the operation simply subtracts $2$. Near zero, the behavior is:\n\n$$1\\to1,\\qquad 0\\to2\\to0.$$\n\nSo:\n\n- Every odd value eventually becomes $1$ permanently.\n- Every even value eventually alternates between $0$ and $2$.\n\nAll odd elements therefore end up equal. The even elements need a closer look.\n\n### Even values have two different timings\n\nWrite an even starting value as $x=2k$. It first reaches $0$ after exactly $k$ operations. After that, it alternates between $0$ and $2$.\n\nAt any sufficiently late common time $T$, its value is:\n\n- $0$ if $T-k$ is even;\n- $2$ if $T-k$ is odd.\n\nThus, two even elements eventually match exactly when their values of $k=x/2$ have the same parity. Equivalently, their original values have the same remainder modulo $4$.\n\nThis gives three groups:\n\n| Original values | Behavior after enough operations |\n| --- | --- |\n| Odd | All equal to $1$ |\n| Remainder $0$ modulo $4$ | All equal to one of $0$ and $2$ |\n| Remainder $2$ modulo $4$ | All equal to the other of $0$ and $2$ |\n\nThe two even groups swap values every operation, but they never overlap. Same dance, opposite feet.\n\n### Why waiting is enough\n\nCould an earlier moment produce a larger frequency?\n\nNo. **Once elements become equal, they remain equal forever:** applying the same function $f(x)=|x-2|$ to equal values gives equal results.\n\nTherefore, any group of equal elements at an earlier time must still be equal at a sufficiently late time. Waiting cannot reduce the maximum frequency.\n\nOnce every element has reached its fixed point or cycle, the three groups above each have a distinct value. Their sizes are therefore exactly the frequencies at that time.\n\nLet $c_r$ be the number of original elements whose remainder modulo $4$ is $r$. The answer is\n\n$$\\boxed{\\max(c_1+c_3,\\ c_0,\\ c_2)}.$$\n\n### Algorithm\n\nFor each test case:\n\n1. Initialize four counters to zero.\n2. Read each value $x$ and increment $c_{x\\bmod4}$.\n3. Output $\\max(c_1+c_3,c_0,c_2)$.\n\nThere is no need to store the array or simulate any operations.\n\nFor example, $[6,7,8]$ has one element in each of the three groups. Eventually it becomes $[0,1,2]$ and then $[2,1,0]$, so its answer is $1$, not $2$.\n\n### Correctness argument\n\nChoose a time after every element has reached $1$ or the cycle $0\\leftrightarrow2$.\n\nAll originally odd elements are now $1$. All originally even elements with the same remainder modulo $4$ have the same value, since their arrival times at $0$ have the same parity. The two even remainder groups have opposite arrival-time parities, so one is $0$ while the other is $2$.\n\nHence the maximum frequency at this time is $\\max(c_1+c_3,c_0,c_2)$, making this value achievable. Any equal elements from an earlier time remain equal at this chosen time, so no earlier frequency can exceed it. The same three group sizes persist afterward. Therefore, the algorithm returns the maximum over all possible numbers of operations, including zero.\n\n### Complexity\n\nEach test case takes $O(n)$ time and $O(1)$ extra space. Across all test cases, the running time is $O(\\sum n)$. Ordinary `int` values suffice for both the input values and the counters.", "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 int cnt[4] = {};\n for (int i = 0; i < n; ++i) {\n int x;\n cin >> x;\n ++cnt[x % 4];\n }\n\n cout << max({cnt[0], cnt[2], cnt[1] + cnt[3]}) << '\\n';\n }\n}\n" }, "id": "msg_0adf2f74e001jvx55mIL0kp5JU", "sessionID": "ses_f520d4942ffeX6i2Z0GTu8Gr9y" }, "parts": [ { "type": "step-start", "id": "prt_0adf30795001qhJvn5qNNfY3kx", "sessionID": "ses_f520d4942ffeX6i2Z0GTu8Gr9y", "messageID": "msg_0adf2f74e001jvx55mIL0kp5JU" }, { "type": "tool", "tool": "StructuredOutput", "callID": "call_4lBl74RaHPuuyYQe3PKDc59o", "state": { "status": "completed", "input": { "status": "success", "reason": null, "hints": [ { "order": 1, "content": "Every operation changes **all** elements together. Try following a few small values until their behavior repeats. What happens to $1$, $2$, $3$, and $4$?" }, { "order": 2, "content": "Every odd value eventually reaches $1$ and stays there. Every even value eventually enters the cycle $0\\to2\\to0$. But do all even values reach the same point in that cycle at the same time?" }, { "order": 3, "content": "Compare starting values $2$ and $4$: they never become equal. Now compare $2$ and $6$: they eventually do. For an even starting value $x$, reaching $0$ takes $x/2$ operations. The parity of that arrival time matters." }, { "order": 4, "content": "Split the original elements into three groups: odd values, values divisible by $4$, and values congruent to $2$ modulo $4$. After enough operations, each group consists of equal values, and the three groups have different values." }, { "order": 5, "content": "Once two elements become equal, they stay equal forever because they always receive the same operation. So no earlier moment can beat the maximum frequency after everything reaches its final fixed point or cycle. If $c_r$ counts original values with remainder $r$ modulo $4$, the answer is $\\max(c_1+c_3,c_0,c_2)$." } ], "editorial": "The catch is **“for all indices.”** You cannot stop one element while another catches up. In particular, “all even numbers eventually become equal” is false: $2$ and $4$ keep missing each other forever.\n\n### What happens to one value?\n\nWhile $x\\ge2$, the operation simply subtracts $2$. Near zero, the behavior is:\n\n$$1\\to1,\\qquad 0\\to2\\to0.$$\n\nSo:\n\n- Every odd value eventually becomes $1$ permanently.\n- Every even value eventually alternates between $0$ and $2$.\n\nAll odd elements therefore end up equal. The even elements need a closer look.\n\n### Even values have two different timings\n\nWrite an even starting value as $x=2k$. It first reaches $0$ after exactly $k$ operations. After that, it alternates between $0$ and $2$.\n\nAt any sufficiently late common time $T$, its value is:\n\n- $0$ if $T-k$ is even;\n- $2$ if $T-k$ is odd.\n\nThus, two even elements eventually match exactly when their values of $k=x/2$ have the same parity. Equivalently, their original values have the same remainder modulo $4$.\n\nThis gives three groups:\n\n| Original values | Behavior after enough operations |\n| --- | --- |\n| Odd | All equal to $1$ |\n| Remainder $0$ modulo $4$ | All equal to one of $0$ and $2$ |\n| Remainder $2$ modulo $4$ | All equal to the other of $0$ and $2$ |\n\nThe two even groups swap values every operation, but they never overlap. Same dance, opposite feet.\n\n### Why waiting is enough\n\nCould an earlier moment produce a larger frequency?\n\nNo. **Once elements become equal, they remain equal forever:** applying the same function $f(x)=|x-2|$ to equal values gives equal results.\n\nTherefore, any group of equal elements at an earlier time must still be equal at a sufficiently late time. Waiting cannot reduce the maximum frequency.\n\nOnce every element has reached its fixed point or cycle, the three groups above each have a distinct value. Their sizes are therefore exactly the frequencies at that time.\n\nLet $c_r$ be the number of original elements whose remainder modulo $4$ is $r$. The answer is\n\n$$\\boxed{\\max(c_1+c_3,\\ c_0,\\ c_2)}.$$\n\n### Algorithm\n\nFor each test case:\n\n1. Initialize four counters to zero.\n2. Read each value $x$ and increment $c_{x\\bmod4}$.\n3. Output $\\max(c_1+c_3,c_0,c_2)$.\n\nThere is no need to store the array or simulate any operations.\n\nFor example, $[6,7,8]$ has one element in each of the three groups. Eventually it becomes $[0,1,2]$ and then $[2,1,0]$, so its answer is $1$, not $2$.\n\n### Correctness argument\n\nChoose a time after every element has reached $1$ or the cycle $0\\leftrightarrow2$.\n\nAll originally odd elements are now $1$. All originally even elements with the same remainder modulo $4$ have the same value, since their arrival times at $0$ have the same parity. The two even remainder groups have opposite arrival-time parities, so one is $0$ while the other is $2$.\n\nHence the maximum frequency at this time is $\\max(c_1+c_3,c_0,c_2)$, making this value achievable. Any equal elements from an earlier time remain equal at this chosen time, so no earlier frequency can exceed it. The same three group sizes persist afterward. Therefore, the algorithm returns the maximum over all possible numbers of operations, including zero.\n\n### Complexity\n\nEach test case takes $O(n)$ time and $O(1)$ extra space. Across all test cases, the running time is $O(\\sum n)$. Ordinary `int` values suffice for both the input values and the counters.", "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 int cnt[4] = {};\n for (int i = 0; i < n; ++i) {\n int x;\n cin >> x;\n ++cnt[x % 4];\n }\n\n cout << max({cnt[0], cnt[2], cnt[1] + cnt[3]}) << '\\n';\n }\n}\n" }, "output": "Structured output captured successfully.", "metadata": { "valid": true }, "title": "Structured Output", "time": { "start": 1789624830253, "end": 1789624830255 } }, "metadata": { "openai": { "itemId": "fc_079d8d5f322fa0d8016aab81cc77d087d1a0c49cda76a56d84" } }, "id": "prt_0adf30796001Ob1qCaZli3ZVeY", "sessionID": "ses_f520d4942ffeX6i2Z0GTu8Gr9y", "messageID": "msg_0adf2f74e001jvx55mIL0kp5JU" }, { "reason": "tool-calls", "type": "step-finish", "tokens": { "total": 23227, "input": 535, "output": 1444, "reasoning": 0, "cache": { "write": 0, "read": 21248 } }, "cost": 0, "id": "prt_0adf3c987001zb9GgwhW3cWXJj", "sessionID": "ses_f520d4942ffeX6i2Z0GTu8Gr9y", "messageID": "msg_0adf2f74e001jvx55mIL0kp5JU" } ] } ] }