{ "info": { "id": "ses_f5326ca6affe1PlflpofN935LJ", "slug": "gentle-comet", "projectID": "global", "directory": "/private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-wKvvnR", "path": "private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-wKvvnR", "title": "Nudge content generation", "agent": "nudge-generation", "model": { "id": "gpt-6-astra", "providerID": "openai", "variant": "max" }, "version": "1.18.30", "summary": { "additions": 0, "deletions": 0, "files": 0 }, "cost": 0, "tokens": { "input": 33656, "output": 4338, "reasoning": 5730, "cache": { "read": 187776, "write": 0 } }, "time": { "created": 1789606311317, "updated": 1789606644694 } }, "messages": [ { "info": { "role": "user", "time": { "created": 1789606311336 }, "agent": "nudge-generation", "model": { "providerID": "openai", "modelID": "gpt-6-astra", "variant": "max" }, "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_0acd935a80017b2mWSKcEoD3hb", "sessionID": "ses_f5326ca6affe1PlflpofN935LJ" }, "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 2264D: \"Dr. Agos's Dark Mode\".\n\nKnown problem metadata:\n- Contest ID: 2264\n- Index: D\n- Name: Dr. Agos's Dark Mode\n- Rating: unknown / unrated\n- Tags: brute force, constructive algorithms, math, strings\n- Problem URL: https://codeforces.com/contest/2264/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. Dr. Agos's Dark Mode
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

Someone has chopped up Dr. Agos's chickens. Furious, he turns to the one thing that still brings him comfort: his OLED screen. Today, he wants almost all of it black.

The screen has a single row of $$$n$$$ pixels. Its pattern is represented by a binary string $$$s$$$; $$$s_i = 0$$$ means that the $$$i$$$-th pixel is dark, and $$$s_i = 1$$$ means that it is lit.

Dr. Agos finds a contiguous segment of pixels irritating if, when read from left to right as a binary integer, its value is divisible by $$$3$$$. More precisely, the segment from position $$$l$$$ to position $$$r$$$ ($$$1 \\le l \\le r \\le n$$$) has value $$$$$$ \\sum_{i=l}^{r} s_i \\cdot 2^{r-i}. $$$$$$ Leading zeroes are allowed. A segment consisting only of dark pixels has value $$$0$$$, which is also divisible by $$$3$$$.

Let $$$f(s)$$$ be the number of irritating segments. Segments with different pairs of endpoints $$$(l,r)$$$ are counted separately, even if their pixel patterns are identical.

Dr. Agos wants a pattern with at most three lit pixels that minimizes $$$f(s)$$$ among all binary strings of length $$$n$$$, including strings with more than three ones. Help him construct such a pattern.

We can show that an answer always exists.

Input

Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \\le t \\le 10^4$$$). The description of the test cases follows.

The only line of each test case contains a single integer $$$n$$$ ($$$1 \\le n \\le 2 \\cdot 10^5$$$) — the number of pixels in the row.

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

Output

For each test case, print a binary string $$$s$$$ of length $$$n$$$ describing Dr. Agos's screen. It must contain at most three ones and minimize $$$f(s)$$$ among all binary strings of length $$$n$$$.

If there are multiple solutions, print any of them.

Example
Input
6
1
2
3
4
5
6
Output
1\n11\n101\n0101\n10101\n010100\n
Note

For $$$n=1$$$, Dr. Agos can light the only pixel, giving $$$s=\\mathtt{1}$$$. The only segment has value $$$1$$$, so it is not irritating and $$$f(s)=0$$$.

For $$$n=2$$$, he can light both pixels, giving $$$s=\\mathtt{11}$$$. The whole row has binary value $$$3$$$ and is irritating, while each individual pixel has value $$$1$$$. Thus, $$$f(s)=1$$$. Every binary string of length $$$2$$$ has at least one irritating segment, so this is optimal.

The displayed patterns are not necessarily unique.

\n\n\n\nSource lookup status:\n\n- Tutorial (en): https://codeforces.com/blog/entry/156680 loaded (200 OK; title: \"Codeforces Round 1121 (Div. 2) Editorial - Codeforces\"; mentions 2264D)\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_0acd935a9001Mr7rBAGaZIlinS", "sessionID": "ses_f5326ca6affe1PlflpofN935LJ", "messageID": "msg_0acd935a80017b2mWSKcEoD3hb" } ] }, { "info": { "parentID": "msg_0acd935a80017b2mWSKcEoD3hb", "role": "assistant", "mode": "nudge-generation", "agent": "nudge-generation", "variant": "max", "path": { "cwd": "/private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-wKvvnR", "root": "/" }, "cost": 0, "tokens": { "total": 8727, "input": 8691, "output": 36, "reasoning": 0, "cache": { "write": 0, "read": 0 } }, "modelID": "gpt-6-astra", "providerID": "openai", "time": { "created": 1789606311466, "completed": 1789606316258 }, "finish": "tool-calls", "id": "msg_0acd9362a0016b7bN0bi4Mvj4w", "sessionID": "ses_f5326ca6affe1PlflpofN935LJ" }, "parts": [ { "type": "step-start", "id": "prt_0acd9407c0010275MApAmLf0Iu", "sessionID": "ses_f5326ca6affe1PlflpofN935LJ", "messageID": "msg_0acd9362a0016b7bN0bi4Mvj4w" }, { "type": "tool", "tool": "webfetch", "callID": "call_vGOvlG6WbkloFGdqwkiToL97", "state": { "status": "completed", "input": { "url": "https://codeforces.com/blog/entry/156680", "format": "markdown", "timeout": 60 }, "output": "---\nimage: https://codeforces.com/codeforces.org/s/21069/images/codeforces-telegram-square-1024x1024.png\ntitle: Codeforces Round 1121 (Div. 2) Editorial - Codeforces\ndescription: Codeforces. Programming competitions and contests, programming community\n---\n\n \n\n \n[![Codeforces](//codeforces.com/codeforces.org/s/21069/images/codeforces-logo-with-telegram-555x130.png \"Codeforces\")](https://codeforces.com/) \n\n[![In English](//codeforces.com/codeforces.org/s/21069/images/flags/24/gb.png \"In English\")](https://codeforces.com/blog/entry/156680?locale=en) [![По-русски](//codeforces.com/codeforces.org/s/21069/images/flags/24/ru.png \"По-русски\")](https://codeforces.com/blog/entry/156680?locale=ru) \n\n[Enter](https://codeforces.com/enter?back=%2Fblog%2Fentry%2F156680)| [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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\"Newbie maruf004\") → [Will Meta Hacker Cup 2026 Happen?](https://codeforces.com/blog/entry/156789) ![New comment\\(s\\)](//codeforces.com/codeforces.org/s/21069/images/icons/comment-12x12.png \"New comment(s)\")\n\n| | [Detailed →](https://codeforces.com/recent-actions) |\n| | --------------------------------------------------- |\n\n* [qwexd](https://codeforces.com/profile/qwexd)\n* [Blog](https://codeforces.com/blog/qwexd)\n* [Teams](https://codeforces.com/teams/with/qwexd)\n* [Submissions](https://codeforces.com/submissions/qwexd)\n* [Groups](https://codeforces.com/groups/with/qwexd)\n* [Contests](https://codeforces.com/contests/with/qwexd)\n* [Problemsetting](https://codeforces.com/contests/writer/qwexd)\n\n### [qwexd's blog](https://codeforces.com/blog/qwexd)\n\n[ Codeforces Round 1121 (Div. 2) Editorial ](https://codeforces.com/blog/entry/156680) \n\nBy [qwexd](https://codeforces.com/profile/qwexd \"Master qwexd\"), 4 days ago, ![In English](//codeforces.com/codeforces.org/s/21069/images/flags/24/gb.png \"In English\") \n\n# Codeforces Round 1121 — Editorial\n\nThanks for participating!\n\n### [2264A - Rumb Needs a Hand](https://codeforces.com/contest/2264/problem/A \"Codeforces Round 1121 (Div. 2)\")\n\nProblem by [qwexd](https://codeforces.com/profile/qwexd \"Master qwexd\").\n\n**Hint**\n\nLook at the elements that are not in the right position. Which of them must be selected? Can selecting any other element help?\n\n**Solution**\n\nEvery element that is not in the right position has to be selected; otherwise, it never moves.\n\nSelecting an element that is already in the right position cannot help. For it to stay correct after the reversal, it has to be paired with itself, because its value occurs nowhere else in the permutation. Thus it can only be the middle selected element, and removing it changes nothing.\n\nSo, if the array is already sorted, choose any one element. Otherwise, select exactly the misplaced elements and reverse them. The answer is `YES` if this sorts the array, and `NO` otherwise.\n\nIf the misplaced positions are listed from left to right, the reversal pairs the first with the last, the second with the second-last, and so on. Check that every pair contains the two values those positions need.\n\nThis takes $$$O(n)$$$ time and $$$O(n)$$$ memory.\n\n**Code (GNU C++17)**\n\n```\n#include \n\nusing namespace std;\n\nint main() {\n ios::sync_with_stdio(false);\n cin.tie(nullptr);\n\n int t;\n cin >> t;\n while (t--) {\n int n;\n cin >> n;\n\n vector p(n);\n for (int &x : p) cin >> x;\n\n vector bad;\n for (int i = 0; i < n; ++i) {\n if (p[i] != i + 1) {\n bad.push_back(i);\n }\n }\n\n bool ok = true;\n int k = bad.size();\n for (int i = 0; i < k; ++i) {\n if (p[bad[i]] != bad[k - 1 - i] + 1) {\n ok = false;\n break;\n }\n }\n\n cout << (ok ? \"YES\\n\" : \"NO\\n\");\n }\n}\n\n```\n\n### [2264B - Knife's Pill Farm](https://codeforces.com/contest/2264/problem/B \"Codeforces Round 1121 (Div. 2)\")\n\nProblem by [qwexd](https://codeforces.com/profile/qwexd \"Master qwexd\").\n\n**Hint 1**\n\nWrite out the score for a few terms and collect the coefficient of each chosen value.\n\n**Hint 2**\n\nFix the last chosen position. Which $$$m-1$$$ earlier values are best, and how can we maintain them while moving this position to the right?\n\n**Solution**\n\nExpanding the score gives\n\n$$$ b\\_1+2(b\\_2-b\\_1)+\\\\cdots+m(b\\_m-b\\_{m-1}) =mb\\_m-\\\\sum\\\\limits\\_{i=1}^{m-1}b\\_i. $$$\n\nOnly the last chosen value has a positive coefficient. Once we decide that $$$a\\_j$$$ is last, we should take the $$$m-1$$$ smallest values before it. Their positions are already in the right order, so they always form a valid subsequence together with $$$a\\_j$$$.\n\nScan $$$j$$$ from left to right. Before considering $$$a\\_j$$$, keep the $$$m-1$$$ smallest values among $$$a\\_1,\\\\ldots,a\\_{j-1}$$$ in a max-heap, and let their sum be $$$s$$$. The best score ending at $$$j$$$ is\n\n$$$ma\\_j-s.$$$\n\nThen insert $$$a\\_j$$$ for later positions and remove the maximum if the heap becomes too large. Take the best score over all $$$j$$$. For $$$m=1$$$, the heap is empty and this simply takes the maximum element.\n\nThe complexity is $$$O(n\\\\log(m+1))$$$ time and $$$O(m)$$$ extra memory.\n\n**Code (GNU C++17)**\n\n```\n#include \n\nusing namespace std;\n\nusing ll = long long;\n\nint main() {\n ios::sync_with_stdio(false);\n cin.tie(nullptr);\n\n int t;\n cin >> t;\n while (t--) {\n int n, m;\n cin >> n >> m;\n\n vector a(n);\n for (ll &x : a) cin >> x;\n\n priority_queue pq;\n ll sum = 0, ans = LLONG_MIN;\n\n for (int i = 0; i < n; ++i) {\n if ((int)pq.size() == m - 1) {\n ans = max(ans, m * a[i] - sum);\n }\n\n pq.push(a[i]);\n sum += a[i];\n if ((int)pq.size() == m) {\n sum -= pq.top();\n pq.pop();\n }\n }\n\n cout << ans << '\\n';\n }\n}\n\n```\n\n### [2264C - Madamant's Skating Dynasty](https://codeforces.com/contest/2264/problem/C \"Codeforces Round 1121 (Div. 2)\")\n\nProblem by [qwexd](https://codeforces.com/profile/qwexd \"Master qwexd\").\n\n**Hint 1**\n\nAfter sorting, every parent lies to the right of their child. What does that imply about cycles and the root?\n\n**Hint 2**\n\nThe total cost is a sum of edge costs. Fix one possible edge; how many choices remain for the other children?\n\n**Solution**\n\nSort the ratings:\n\n$$$ b\\_0 \\\\lt b\\_1 \\\\lt \\\\cdots \\\\lt b\\_{n-1}. $$$\n\nThe maximum is forced to be the root. Give every other position $$$i$$$ any parent $$$j \\\\gt i$$$. Following parent links always moves right, so there is no cycle and every path ends at the maximum. Thus any such choices form a valid tree.\n\nWe cannot enumerate the trees. Instead, consider one possible edge at a time and add its cost once for every tree containing it.\n\nFix the edge from child $$$i$$$ to parent $$$j \\\\gt i$$$. Its cost is $$$b\\_j-b\\_i$$$. Every other child $$$k$$$ still has $$$n-k-1$$$ choices, so this edge appears in\n\n$$$ W\\_i=\\\\prod\\\\limits\\_{\\\\substack{0\\\\le k \\\\lt n-1\\\\\\\\k\\\\ne i}}(n-k-1) $$$\n\ntrees. The value $$$W\\_i$$$ does not depend on $$$j$$$, so all edges leaving $$$i$$$ contribute\n\n$$$ W\\_i\\\\sum\\\\limits\\_{j=i+1}^{n-1}(b\\_j-b\\_i) =W\\_i\\\\left(\\\\sum\\\\limits\\_{j=i+1}^{n-1}b\\_j\\\\right) - W\\_i (n-i-1)b\\_i. $$$\n\nA suffix sum gives the sum of ratings to the right of each $$$i$$$. Prefix and suffix products of the numbers $$$n-k-1$$$ give every $$$W\\_i$$$ without division.\n\nThe complexity is $$$O(n\\\\log n)$$$ time and $$$O(n)$$$ memory.\n\n**Code (GNU C++17)**\n\n```\n#include \n\nusing namespace std;\n\nusing ll = long long;\n\nconst int MOD = 998244353;\n\nint main() {\n ios::sync_with_stdio(false);\n cin.tie(nullptr);\n\n int t;\n cin >> t;\n while (t--) {\n int n;\n cin >> n;\n vector b(n);\n for (ll &x : b) cin >> x;\n sort(b.begin(), b.end());\n\n vector sum(n + 1);\n for (int i = n - 1; i >= 0; --i) {\n sum[i] = sum[i + 1] + b[i];\n }\n\n vector pref(n, 1), suff(n, 1);\n for (int i = 0; i < n - 1; ++i) {\n pref[i + 1] = pref[i] * (n - i - 1) % MOD;\n }\n for (int i = n - 2; i >= 0; --i) {\n suff[i] = suff[i + 1] * (n - i - 1) % MOD;\n }\n\n ll ans = 0;\n for (int i = 0; i < n - 1; ++i) {\n ll diff = sum[i + 1] - (n - i - 1) * b[i];\n ll ways = pref[i] * suff[i + 1] % MOD;\n ans = (ans + diff % MOD * ways) % MOD;\n }\n cout << ans << '\\n';\n }\n}\n\n```\n\n### [2264D - Dr. Agos's Dark Mode](https://codeforces.com/contest/2264/problem/D \"Codeforces Round 1121 (Div. 2)\")\n\nProblem by [qwexd](https://codeforces.com/profile/qwexd \"Master qwexd\").\n\n**Hint 1**\n\nSince $$$2\\\\equiv-1\\\\pmod 3$$$, try an alternating prefix sum.\n\n**Hint 2**\n\nA divisible substring corresponds to two equal prefix remainders. How should the counts of the three remainders be distributed?\n\n**Hint 3**\n\nWith three ones, the prefix remainder is constant on four blocks. Try their boundaries around $$$n/3$$$, $$$2n/3$$$, and the end of the string.\n\n**Solution**\n\nModulo $$$3$$$, the powers of $$$2$$$ alternate between $$$1$$$ and $$$-1$$$. Define the alternating prefix sum\n\n$$$ p\\_0=0,\\\\qquad p\\_i\\\\equiv\\\\sum\\\\limits\\_{j=1}^{i}(-1)^js\\_j\\\\pmod 3\\. $$$\n\nIf $$$v$$$ is the binary value of $$$s\\_l\\\\ldots s\\_r$$$, then\n\n$$$ v\\\\equiv(-1)^r(p\\_r-p\\_{l-1})\\\\pmod 3\\. $$$\n\nTherefore, this substring is divisible by $$$3$$$ exactly when $$$p\\_{l-1}=p\\_r$$$.\n\nSuppose the remainders $$$0,1,2$$$ occur $$$c\\_0,c\\_1,c\\_2$$$ times among the $$$n+1$$$ prefix sums. Every pair of equal prefix remainders gives one divisible substring, so\n\n$$$ f(s)=\\\\binom{c\\_0}{2}+\\\\binom{c\\_1}{2}+\\\\binom{c\\_2}{2}. $$$\n\nFor a fixed sum $$$c\\_0+c\\_1+c\\_2=n+1$$$, this is smallest when the three counts differ by at most one.\n\nNow we only need to build a string with such balanced counts. A zero repeats the current prefix remainder. A one changes it by $$$-1$$$ at an odd position and by $$$+1$$$ at an even position. Thus, three ones divide the prefix sums into four constant blocks. To make the three remainders appear almost equally often, the block boundaries should be near the thirds of the string.\n\nFor $$$n\\\\ge3$$$, simply try:\n\n* the first one at $$$\\\\lfloor n/3\\\\rfloor$$$ or $$$\\\\lfloor n/3\\\\rfloor+1$$$;\n* the second one at $$$\\\\lfloor2n/3\\\\rfloor$$$ or $$$\\\\lfloor2n/3\\\\rfloor+1$$$;\n* either no third one, or a third one at $$$n$$$.\n\nFor each of these at most eight strings, count how many prefix sums have each remainder. Output the first one whose three counts differ by at most one.\n\nThe searched ranges contain the following construction. Let $$$q=\\\\lfloor n/3\\\\rfloor$$$ and consider a string of length $$$3q+2$$$.\n\n* If $$$q$$$ is odd, put ones at $$$q$$$, $$$2q+1$$$, and $$$3q+2$$$. The four blocks have remainders $$$0,2,1,0$$$ and lengths $$$q,q+1,q+1,1$$$.\n* If $$$q$$$ is even, put ones at $$$q+1$$$, $$$2q+1$$$, and $$$3q+2$$$. The blocks have remainders $$$0,2,1,2$$$ and lengths $$$q+1,q,q+1,1$$$.\n\nIn both cases, each remainder occurs $$$q+1$$$ times. The real length $$$n$$$ is obtained by cutting off at most two characters. The last two prefix remainders are different, so the remaining counts are still balanced. If position $$$3q+2$$$ was cut off, this is exactly the choice with no third one. Hence, the string is among the eight candidates above.\n\nFor $$$n=1$$$ and $$$n=2$$$, output `1` and `10`, respectively. For $$$n=2$$$, three distinct prefix remainders are impossible, while `10` has only one divisible substring, so it is optimal.\n\nThe complexity is $$$O(n)$$$ time and $$$O(n)$$$ memory per test case.\n\n**Code (GNU C++17)**\n\n```\n#include \n\nusing namespace std;\n\nbool hasBalancedPrefixes(const string& s) {\n array count{1, 0, 0};\n int remainder = 0;\n\n for (int i = 0; i < (int)s.size(); ++i) {\n if (s[i] == '1') {\n remainder = (remainder + (i % 2 == 0 ? 2 : 1)) % 3;\n }\n ++count[remainder];\n }\n\n return *max_element(count.begin(), count.end())\n - *min_element(count.begin(), count.end()) <= 1;\n}\n\nstring constructString(int n) {\n if (n == 1) {\n return \"1\";\n }\n if (n == 2) {\n return \"10\";\n }\n\n for (int first = n / 3; first <= n / 3 + 1; ++first) {\n for (int second = 2 * n / 3; second <= 2 * n / 3 + 1; ++second) {\n for (int putLastOne = 0; putLastOne <= 1; ++putLastOne) {\n if (first >= second || (putLastOne && second >= n)) {\n continue;\n }\n\n string s(n, '0');\n s[first - 1] = '1';\n s[second - 1] = '1';\n if (putLastOne) {\n s[n - 1] = '1';\n }\n\n if (hasBalancedPrefixes(s)) {\n return s;\n }\n }\n }\n }\n\n assert(false);\n return {};\n}\n\nint main() {\n ios::sync_with_stdio(false);\n cin.tie(nullptr);\n\n int t;\n cin >> t;\n while (t--) {\n int n;\n cin >> n;\n cout << constructString(n) << '\\n';\n }\n return 0;\n}\n\n```\n\n### [2264E1 - A Prime Flood (Easy Version)](https://codeforces.com/contest/2264/problem/E1 \"Codeforces Round 1121 (Div. 2)\")\n\nProblem by [qwexd](https://codeforces.com/profile/qwexd \"Master qwexd\").\n\n**Hint 1**\n\nAn operation never changes the order of two values. If the minimum and maximum become equal, what happens to every value between them?\n\n**Hint 2**\n\nLet $$$dp\\[x\\]\\[y\\]$$$ be the answer for minimum $$$x$$$ and maximum $$$y$$$. For $$$x \\\\lt y$$$, can we decrease $$$y$$$ while leaving $$$x$$$ unchanged?\n\n**Hint 3**\n\nAfter computing every $$$dp\\[x\\]\\[y\\]$$$, count subsequences with minimum $$$x$$$ and maximum $$$y$$$.\n\n**Solution**\n\nFirst ignore the counting of subsequences. Let $$$dp\\[x\\]\\[y\\]$$$ be the largest value to which the pair $$$\\[x,y\\]$$$ can be made equal.\n\nEvery operation preserves the order of the values. Therefore, if the minimum and maximum of a subsequence become equal, every value between them becomes equal as well. The contribution of a subsequence $$$b$$$ is simply\n\n$$$ dp\\[\\\\min(b)\\]\\[\\\\max(b)\\]. $$$\n\nWe can compute this table directly. Clearly, $$$dp\\[x\\]\\[x\\]=x$$$. Now suppose $$$x \\\\lt y$$$ and try to decrease the larger endpoint.\n\nIf some prime divides $$$y$$$ but does not divide $$$x$$$, choosing it changes the pair to $$$\\[x,y-1\\]$$$. Also, any sequence that equalizes $$$x$$$ and $$$y$$$ must equalize the value $$$y-1$$$ between them. Thus, this case gives $$$dp\\[x\\]\\[y-1\\]$$$.\n\nOtherwise, every prime divisor of $$$y$$$ also divides $$$x$$$, so choosing any of them changes the pair to $$$\\[x-1,y-1\\]$$$.\n\nWhy can doing something else first not give a better result? Look at the first operation in any successful sequence that decreases $$$y$$$. Immediately after it, the upper endpoint is $$$y-1$$$, while the lower endpoint is already at most $$$x-1$$$: it either decreased earlier, or the same prime decreases it together with $$$y$$$. The remaining operations equalize these two values, so order preservation means that they also equalize the value $$$x-1$$$ between them. Therefore, this case gives $$$dp\\[x-1\\]\\[y-1\\]$$$.\n\nThe transition is\n\n$$$ dp\\[x\\]\\[y\\]= \\\\begin{cases} dp\\[x-1\\]\\[y-1\\],&\\\\text{if every prime divisor of }y\\\\text{ divides }x,\\\\\\\\ dp\\[x\\]\\[y-1\\],&\\\\text{otherwise}. \\\\end{cases} $$$\n\nHere the limit is only $$$M=3\\\\,000$$$, so we can build the whole $$$M\\\\times M$$$ table. Let `need[y]` be the product of the distinct prime divisors of $$$y$$$. Sieve all `need[y]`; the first case applies exactly when `x % need[y] == 0`. Process $$$y-x$$$ from small to large and $$$x$$$ from small to large, so both states used by the transition are ready.\n\nNow count subsequences by their minimum and maximum. Let $$$c\\_v$$$ be the number of occurrences of $$$v$$$.\n\nIf both endpoints are $$$v$$$, any nonempty subset of its occurrences works, giving $$$2^{c\\_v}-1$$$ choices.\n\nFor $$$x \\\\lt y$$$, choose at least one occurrence of both $$$x$$$ and $$$y$$$. Every occurrence with value strictly between them is optional, so the number of choices is\n\n$$$ (2^{c\\_x}-1)(2^{c\\_y}-1) 2^{\\\\sum\\\\limits\\_{x \\\\lt v \\\\lt y}c\\_v}. $$$\n\nUse prefix sums of the frequencies for the exponent, multiply each count by $$$dp\\[x\\]\\[y\\]$$$, and add everything to the answer.\n\nThe one-time DP takes $$$O(M^2)$$$ time and memory. Each test case takes $$$O(n^2)$$$ time and $$$O(n)$$$ additional memory.\n\n**Code (GNU C++17)**\n\n```\n#include \n\nusing namespace std;\n\nconst int MAXV = 3000;\nconst int MOD = 998244353;\n\nint dp[MAXV + 1][MAXV + 1], need[MAXV + 1], p2[MAXV + 1];\n\nvoid precompute() {\n fill(need, need + MAXV + 1, 1);\n for (int p = 2; p <= MAXV; ++p) {\n if (need[p] != 1) {\n continue;\n }\n for (int value = p; value <= MAXV; value += p) {\n need[value] *= p;\n }\n }\n\n for (int x = 1; x <= MAXV; ++x) {\n dp[x][x] = x;\n }\n for (int difference = 1; difference < MAXV; ++difference) {\n for (int x = 1; x + difference <= MAXV; ++x) {\n int y = x + difference;\n if (x % need[y] == 0) {\n dp[x][y] = dp[x - 1][y - 1];\n } else {\n dp[x][y] = dp[x][y - 1];\n }\n }\n }\n\n p2[0] = 1;\n for (int i = 1; i <= MAXV; ++i) {\n p2[i] = 2LL * p2[i - 1] % MOD;\n }\n}\n\nint main() {\n ios::sync_with_stdio(false);\n cin.tie(nullptr);\n\n precompute();\n\n int t;\n cin >> t;\n while (t--) {\n int n;\n cin >> n;\n vector freq(n + 1);\n for (int i = 0; i < n; ++i) {\n int value;\n cin >> value;\n ++freq[value];\n }\n\n vector prefix(n + 1);\n for (int value = 1; value <= n; ++value) {\n prefix[value] = prefix[value - 1] + freq[value];\n }\n\n long long answer = 0;\n for (int x = 1; x <= n; ++x) {\n if (freq[x] == 0) {\n continue;\n }\n int chooseX = p2[freq[x]] - 1;\n answer = (answer + 1LL * x * chooseX) % MOD;\n\n for (int y = x + 1; y <= n; ++y) {\n if (freq[y] == 0) {\n continue;\n }\n int chooseY = p2[freq[y]] - 1;\n\n int middle = prefix[y - 1] - prefix[x];\n long long ways = 1LL * chooseX * chooseY % MOD;\n ways = ways * p2[middle] % MOD;\n answer = (answer + ways * dp[x][y]) % MOD;\n }\n }\n cout << answer << '\\n';\n }\n}\n\n```\n\n### [2264E2 - A Prime Flood (Hard Version)](https://codeforces.com/contest/2264/problem/E2 \"Codeforces Round 1121 (Div. 2)\")\n\nProblem by [qwexd](https://codeforces.com/profile/qwexd \"Master qwexd\") and solved for bigger constraints by [jeroenodb](https://codeforces.com/profile/jeroenodb \"Legendary Grandmaster jeroenodb\").\n\n**Hint 1**\n\nLet $$$dp\\[x\\]\\[y\\]$$$ be the answer for minimum $$$x$$$ and maximum $$$y$$$. E1 builds and sums over every pair. Fix $$$x$$$ and scan larger values of $$$y$$$. When can $$$dp\\[x\\]\\[y\\]$$$ differ from $$$dp\\[x\\]\\[y-1\\]$$$?\n\n**Hint 2**\n\nLet $$$r(y)$$$ be the product of the distinct prime divisors of $$$y$$$. The row for $$$x$$$ can change only at positions where $$$r(y)\\\\mid x$$$. Generate these positions by fixing $$$y$$$ and visiting the smaller multiples of $$$r(y)$$$.\n\n**Hint 3**\n\nFor fixed $$$x$$$, separate the number of subsequences with endpoints $$$(x,y)$$$ into an $$$x$$$-only factor and a $$$y$$$-only factor. Prefix sums of the $$$y$$$-only factors can add a whole constant part of the row at once.\n\n**Solution**\n\nThe easy solution builds the full DP table and then loops over every minimum and maximum. We need to compress both quadratic parts.\n\nLet $$$r(y)$$$ be the product of the distinct prime divisors of $$$y$$$. The transition from E1 is\n\n$$$ dp\\[x\\]\\[y\\]= \\\\begin{cases} dp\\[x-1\\]\\[y-1\\],&r(y)\\\\mid x,\\\\\\\\ dp\\[x\\]\\[y-1\\],&r(y)\\\\nmid x. \\\\end{cases} $$$\n\nFix $$$x$$$ and read the row from left to right. Unless $$$r(y)$$$ divides $$$x$$$, the value is copied from the previous position. Thus, the row can change only at positions satisfying $$$r(y)\\\\mid x$$$; call them blockers.\n\nFor example, the blockers of row $$$2$$$ are $$$4,8,16,\\\\ldots$$$. At $$$y=4$$$, the row changes from $$$2$$$ to $$$dp\\[1\\]\\[3\\]=1$$$. Every later blocker also reads $$$1$$$ from row $$$1$$$, so the whole row is stored as only $$$(2,2)$$$ and $$$(4,1)$$$.\n\nWe can generate all blockers without checking every pair. For each $$$y$$$, visit\n\n$$$ x=r(y),2r(y),3r(y),\\\\ldots\\\\qquad(x \\\\lt y) $$$\n\nand add $$$y$$$ to the blocker list of each visited $$$x$$$. Processing $$$y$$$ in increasing order keeps every list sorted.\n\nNow build the rows in increasing order of $$$x$$$. Store a row as breakpoints $$$(s,z)$$$: its value becomes $$$z$$$ starting at position $$$s$$$. Begin row $$$x$$$ with $$$(x,x)$$$. At a blocker $$$y$$$, advance a pointer in row $$$x-1$$$ while the next breakpoint starts at or before $$$y-1$$$. Its current value is $$$dp\\[x-1\\]\\[y-1\\]$$$. Add a new breakpoint only when this value differs from the previous one.\n\nFor the fixed limit $$$M=300\\\\,000$$$, generating the blocker lists visits $$$5\\\\,970\\\\,168$$$ pairs. Unfortunately, a bound on the number of generated pairs is not easy to calculate for general $$$n$$$, so the best way to solve the problem in-contest is to experimentally verify that the number of pairs is small enough for the given $$$n$$$. After equal neighboring parts are merged, all rows together contain $$$640\\\\,704$$$ breakpoints, and no row contains more than eight.\n\nWe still need to avoid iterating over every pair of endpoints. Let $$$c\\_v$$$ be the frequency of $$$v$$$, and write\n\n$$$ C\\_v=2^{c\\_v}-1,\\\\qquad S(v)=\\\\sum\\\\limits\\_{u\\\\le v}c\\_u. $$$\n\nFor $$$x \\\\lt y$$$, the number of subsequences with minimum $$$x$$$ and maximum $$$y$$$ is\n\n$$$ C\\_xC\\_y2^{S(y-1)-S(x)} =\\\\left(C\\_x2^{-S(x)}\\\\right) \\\\left(C\\_y2^{S(y-1)}\\\\right). $$$\n\nCall the two factors $$$L\\_x$$$ and $$$R\\_y$$$. Build prefix sums of $$$R\\_y$$$. Intersect each compressed part with $$$x \\\\lt y\\\\le n$$$. If the remaining interval is $$$u\\\\le y\\\\le v$$$ and the row value there is $$$z$$$, its whole contribution is\n\n$$$ L\\_xz\\\\sum\\\\limits\\_{y=u}^{v}R\\_y, $$$\n\nwhich is one prefix-sum query. Skip values with $$$c\\_x=0$$$. The case $$$x=y$$$ contributes $$$xC\\_x$$$ separately.\n\nThe radical sieve takes $$$O(M\\\\log\\\\log M)$$$ time. The remaining preprocessing is linear in the $$$5\\\\,970\\\\,168$$$ generated pairs and $$$640\\\\,704$$$ stored breakpoints. Afterward, each present minimum scans at most eight parts, so the additional work is virtually linear in the total input size. The blocker lists and compressed rows have the sizes given above; together with the $$$O(M)$$$ arrays, they fit comfortably in memory.\n\n**Code (GNU C++17)**\n\n```\n#include \n\nusing namespace std;\n\nconst int MAXN = 300000;\nconst int MOD = 998244353;\n\nstruct Breakpoint {\n int pos;\n int value;\n};\n\narray freq, prefixCount, prefix, p2, ip2;\nvector> rows;\n\nvoid buildRows() {\n vector rad(MAXN + 1, 1);\n for (int p = 2; p <= MAXN; ++p) {\n if (rad[p] != 1) {\n continue;\n }\n for (int x = p; x <= MAXN; x += p) {\n rad[x] *= p;\n }\n }\n\n vector> blockers(MAXN + 1);\n for (int y = 2; y <= MAXN; ++y) {\n for (int x = rad[y]; x < y; x += rad[y]) {\n blockers[x].push_back(y);\n }\n }\n\n rows.resize(MAXN + 1);\n rows[0].push_back({0, 0});\n for (int x = 1; x <= MAXN; ++x) {\n auto &row = rows[x];\n auto &previous = rows[x - 1];\n row.push_back({x, x});\n\n int ptr = 0;\n for (int y : blockers[x]) {\n while (ptr + 1 < (int)previous.size() &&\n previous[ptr + 1].pos < y) {\n ++ptr;\n }\n int value = previous[ptr].value;\n if (value != row.back().value) {\n row.push_back({y, value});\n }\n }\n }\n}\n\nvoid buildPowers() {\n const int inv2 = (MOD + 1) / 2;\n p2[0] = ip2[0] = 1;\n for (int i = 1; i <= MAXN; ++i) {\n p2[i] = 2LL * p2[i - 1] % MOD;\n ip2[i] = 1LL * inv2 * ip2[i - 1] % MOD;\n }\n}\n\nint rangeSum(int left, int right) {\n if (left > right) {\n return 0;\n }\n return (prefix[right] - prefix[left - 1] + MOD) % MOD;\n}\n\nint main() {\n ios::sync_with_stdio(false);\n cin.tie(nullptr);\n\n buildRows();\n buildPowers();\n\n int t;\n cin >> t;\n while (t--) {\n int n;\n cin >> n;\n vector values;\n for (int i = 0; i < n; ++i) {\n int x;\n cin >> x;\n if (freq[x] == 0) {\n values.push_back(x);\n }\n ++freq[x];\n }\n\n int before = 0, ans = 0;\n prefixCount[0] = prefix[0] = 0;\n for (int x = 1; x <= n; ++x) {\n int ways = p2[freq[x]] - 1;\n int rightWeight = 1LL * ways * p2[before] % MOD;\n prefix[x] = prefix[x - 1] + rightWeight;\n if (prefix[x] >= MOD) {\n prefix[x] -= MOD;\n }\n ans = (ans + 1LL * x * ways) % MOD;\n before += freq[x];\n prefixCount[x] = before;\n }\n\n for (int x : values) {\n int ways = p2[freq[x]] - 1;\n int leftWeight = 1LL * ways * ip2[prefixCount[x]] % MOD;\n\n int rowSum = 0;\n auto &row = rows[x];\n for (int i = 0; i < (int)row.size(); ++i) {\n int left = max(x + 1, row[i].pos);\n if (left > n) {\n break;\n }\n int right = n;\n if (i + 1 < (int)row.size()) {\n right = min(n, row[i + 1].pos - 1);\n }\n rowSum = (rowSum +\n 1LL * row[i].value * rangeSum(left, right)) % MOD;\n }\n ans = (ans + 1LL * leftWeight * rowSum) % MOD;\n }\n\n cout << ans << '\\n';\n for (int x : values) {\n freq[x] = 0;\n }\n }\n}\n\n```\n\n### [2264F - Deranged Calculator](https://codeforces.com/contest/2264/problem/F \"Codeforces Round 1121 (Div. 2)\")\n\nProblem by [jeroenodb](https://codeforces.com/profile/jeroenodb \"Legendary Grandmaster jeroenodb\"). Testing tool created by [turska](https://codeforces.com/profile/turska \"Master turska\")\n\n**Hint 1**\n\nTry to make an expression for $$$n!$$$ as a warm-up\n\n**Hint 2**\n\nTry to use round and division to make an expression which is $$$0$$$ or $$$1$$$ at specific $$$n$$$.\n\n**Hint 3**\n\nLook for a formula relating $$$n!$$$ and the number of derangements.\n\n**Solution**\n\nNote that $$$10\\\\,000$$$ looks like a lot of characters for an expression, but for $$$n$$$ up to $$$50$$$, due to the particular restrictions of the Deranged Calculator language, this limit is quite tight. So for everything we come up with, we have to be sure we are not wasting too many characters. The model solution uses $$$5969$$$ characters, and there are even better methods that achieve (way) fewer characters.\n\nLet's first try to solve an easier problem: what if, instead of the number of derangements of $$$n$$$ elements, we try to calculate the number of permutations, $$$n!$$$, instead?\n\nWe can immediately notice that if we want to use integer constants in our program, we have to make them using workarounds such as `n/n` or `(n+n+n)/n`. To save some length, we can precompute a dynamic programming table, which tells us a small construction for the constant $$$i$$$ for $$$1\\\\leq i\\\\leq50$$$. Transitions can be coded to use `+` and `*` operators, considering all ways to make $$$i=x+y$$$ or $$$i=x\\\\cdot y$$$, with some appropriate use of parentheses and `/n`. It turns out that if the the other parts of the expression are well optimized, even writing constants naively in the form $$$(n+n+\\\\dots+n)/n$$$ can work. In less optimized expressions, optimizing expressions for constants is maybe required\n\nNext up, we have to face the issue that we cannot use recursion or loops to calculate the factorial. The one saving grace is that the `round` function can help us to make some kind of step functions so we can filter out specific $$$n$$$ for our needs.\n\nIt would be great if we could make the following program:\n\n```\n(1*[n>=2]+1)*(2*[n>=3]+1)*...*(49*[n>=50]+1)\n\n```\n\nIn such a program, we basically compute $$$50!$$$, but we modulate which terms we want to multiply and which we replace with $$$1$$$. Here `[n>=2]` is an indicator expression that should output $$$1$$$ if the given inequality is true and $$$0$$$ otherwise.\n\nSo now our goal has shifted to building an indicator expression `[n>=x]` for a given constant $$$x$$$. Here lots of tricks can be used involving the `round` function. One particularly nice one is the following: `round(n/(n+x))`. For a given constant $$$x$$$ (for which we have a short expression precomputed), this function gives exactly $$$1$$$ once $$$n\\\\geq x$$$, and $$$0$$$ for $$$2\\\\leq n \\\\lt x$$$. So this solves calculating factorials in the DC language in a reasonable number of characters (this depends on how optimized your `round`\\-based \"if statement\" is, and how well optimized the DP is).\n\nBut what about derangements? It turns out there are some nice formulas relating factorials and derangements. One particular fact is that\n\n$$$ \\\\operatorname{round}(n!/e)=D\\_n, $$$\n\nwhere $$$e\\\\approx2.71828$$$ is Euler's number, and $$$D\\_n$$$ is the number of derangements for this $$$n$$$. This means we can basically reuse all our work for the factorial case, but now we have to find a way to get a sufficiently precise approximation of $$$e$$$ in our program. There are again many possibilities, but as we already know how to create all integer constants up to $$$50$$$ with a small number of characters, one way is to use the Taylor expansion of $$$e^x$$$, evaluated at $$$x=1$$$:\n\n$$$ e^1=\\\\sum\\\\limits\\_{k=0}^{\\\\infty}\\\\frac{1}{k!}. $$$\n\nWe can cut off the series at some point, and as we are dividing $$$50!$$$ by $$$e$$$, it makes sense that we need roughly the first $$$50$$$ terms of the expansion, as we need a relative error of $$$\\\\sim \\\\frac{1}{50!}$$$. Summing all the terms from $$$k=0$$$ to $$$k=50$$$ can be confirmed to work. Naively writing a sum of the inverse factorials will use $$$\\\\sim50^2/2$$$ constants, and this will probably not fit. Instead, we can rewrite the sum as\n\n$$$ 2+\\\\frac{1}{2}\\\\left(1+\\\\frac{1}{3}\\\\left(1+\\\\frac{1}{4}(\\\\dots)\\\\right)\\\\right), $$$\n\nmaking use of the common terms in the factorials. This only uses $$$\\\\sim50$$$ constants, and some $$$1$$$'s, which are very cheap.\n\nTaking this all together, we can write our final solution as `round(n!/e)`, with our expressions for $$$n!$$$ and $$$e$$$ substituted.\n\nThe model solution tries to do some more tricks to optimize the precomputation DP for the constants, but they are not needed:\n\n* Use a bigger DP table $$$DP\\_{i,j}$$$, for $$$0\\\\leq j \\\\lt 4$$$, which calculates a small-length expression for getting an expression with value $$$i\\\\cdot n^j$$$, and also consider dividing using `/n` as transitions.\n* Think about when exactly you can skip parentheses.\n* Try to also use the `-` operator in your DP. Due to both `-` and `/` transitions, the DP is no longer acyclic, but as the table is very small, we can just keep iterating through all possible transitions multiple times, and eventually the DP table stays unchanged.\n\nAs a bonus, what is the lowest you can push the number of characters?\n\n**Code (PyPy 3)**\n\n```\nN = 50\nL = 4\ndp = [[None]*L for i in range(N+1)]\ndef upd(l,n,s):\n if not dp[n][l] or len(s)1: return '(' + s + ')'\n return s\n\nfor iter in range(1,N):\n for l1 in range(0,L):\n for l2 in range(0,L):\n for i in range(1,N+1):\n for j in range(1,N+1):\n if dp[i][l1] and dp[j][l2]:\n a = dp[i][l1]\n b = dp[j][l2]\n if i=0: upd(l1-l2, i//j, f\"{parens(a)}/{parens(b)}\")\n\ne = \"n/n\"\nfor i in range(N,0,-1): e = f\"n/n+n/({dp[i][1]})*({e})\"\n\ndef geqn(i): return f\"round(n/({dp[i][0]}+n))\"\n\ndef factorial():\n res = \"n/n\"\n for j in range(2,N+1):\n res += f\"*({geqn(j)}*({dp[j-1][0]})+n/n)\"\n return res\nfact = factorial()\nans = f\"round({fact}/({e}))\"\nprint(ans)\n\n```\n\n**Code (PyPy 3) simplified and golfed**\n\n```\ndef n(x):\n if x==1: return \"n/n\"\n else: return '(' + \"+\".join(\"n\"*x) + ')/n'\nN, e, fact = 50, n(1),n(1)\ndef geqn(i): return f\"round(n/({n(i)}+n))\"\nfor j in range(2,N+1): fact += f\"*({geqn(j)}*({n(j-1)})+n/n)\"\nfor i in range(N,0,-1): e = f\"n/n+n/({'+'.join('n'*i)})*({e})\"\nprint(f\"round({fact}/({e}))\")\n\n```\n\n![](//codeforces.com/codeforces.org/s/21069/images/icons/paperclip-16x16.png) Tutorial of [Codeforces Round 1121 (Div. 2)](https://codeforces.com/contest/2264) \n\n* [![Vote: I like it](//codeforces.com/codeforces.org/s/21069/images/actions/voteup-gray.png \"Vote: I like it\")](#)\n* +133\n* [![Vote: I do not like it](//codeforces.com/codeforces.org/s/21069/images/actions/votedown-gray.png \"Vote: I do not like it\")](#)\n\n* [![Author](//codeforces.com/codeforces.org/s/21069/images/blog/user_16x16.png \"Author\")](https://codeforces.com/profile/qwexd) [qwexd ](https://codeforces.com/profile/qwexd)\n* ![Publication date](//codeforces.com/codeforces.org/s/21069/images/blog/date_16x16.png \"Publication date\") 4 days ago\n* [![Comments](//codeforces.com/codeforces.org/s/21069/images/blog/comments_16x16.png \"Comments\")](https://codeforces.com/blog/entry/156680#comments) [26 ](https://codeforces.com/blog/entry/156680#comments)\n\n \n![Comments](//codeforces.com/codeforces.org/s/21069/images/icons/comments-48x48.png \"Comments\") Comments (26) \n\n[Write comment?](#) \n\n| » [AksLolCoding](https://codeforces.com/profile/AksLolCoding \"Grandmaster AksLolCoding\") | 3 days ago, [show](#) # |\n| ---------------------------------------------------------------------------------------- | ----------------------- |\n\n| » [ ![](https://codeforces.com/userpic.codeforces.org/2948677/avatar/f15474cd5b82d030.jpg) ](https://codeforces.com/profile/AksLolCoding) [AksLolCoding](https://codeforces.com/profile/AksLolCoding \"Grandmaster AksLolCoding\") | 3 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156680?#comment-1392481 \"Link to comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/21069/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) +9 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/21069/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) F can be solved in 1661 charactersSolution by ChatGPT: [390658725](https://codeforces.com/contest/2264/submission/390658725 \"Submission 390658725 by AksLolCoding\") [→](#) [Reply](#) |\n| -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n| » [RedStar\\_KAHA](https://codeforces.com/profile/RedStar%5FKAHA \"Pupil RedStar_KAHA\") | 3 days ago, [show (+1)](#) # |\n| ------------------------------------------------------------------------------------- | ---------------------------- |\n\n| » [ ![](https://codeforces.com/userpic.codeforces.org/6201707/avatar/1424f28bc0f8a6ae.jpg) ](https://codeforces.com/profile/RedStar%5FKAHA) [RedStar\\_KAHA](https://codeforces.com/profile/RedStar%5FKAHA \"Pupil RedStar_KAHA\") | 3 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156680?#comment-1392482 \"Link to comment\") \\| [←](#)Rev. 2 [→](#) [![Vote: I like it](//codeforces.com/codeforces.org/s/21069/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) \\-37 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/21069/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) Deleted [→](#) [Reply](#) |\n| ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n* | » » [ish4an](https://codeforces.com/profile/ish4an \"Expert ish4an\") | 3 days ago, [show](#) # |\n| ------------------------------------------------------------------- | ----------------------- |\n\n| » » [ ![](https://codeforces.com/userpic.codeforces.org/3955333/avatar/b75406e855497b18.jpg) ](https://codeforces.com/profile/ish4an) [ish4an](https://codeforces.com/profile/ish4an \"Expert ish4an\") | 3 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156680?#comment-1392483 \"Link to comment\") [^](#comment-1392482 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/21069/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) +1 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/21069/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) stop spamming ts [→](#) [Reply](#) |\n| ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n* | » » [qwexd](https://codeforces.com/profile/qwexd \"Master qwexd\") | 3 days ago, [show (+1)](#) # |\n| ---------------------------------------------------------------- | ---------------------------- |\n\n\n...51392 bytes truncated...\n\nThe tool call succeeded but the output was truncated. Full output saved to: /Users/zayd/.local/share/opencode/tool-output/tool_0acd948c900135p8Hot96mvkRz\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_0acd948c900135p8Hot96mvkRz" }, "title": "https://codeforces.com/blog/entry/156680 (text/markdown; charset=utf-8)", "time": { "start": 1789606314945, "end": 1789606316242 } }, "metadata": { "openai": { "itemId": "fc_02b302fa3d992c8d016aab39aa0d5487d2871496e3981e059e" } }, "id": "prt_0acd94081001BlMXZ1u7eoV2sz", "sessionID": "ses_f5326ca6affe1PlflpofN935LJ", "messageID": "msg_0acd9362a0016b7bN0bi4Mvj4w" }, { "reason": "tool-calls", "type": "step-finish", "tokens": { "total": 8727, "input": 8691, "output": 36, "reasoning": 0, "cache": { "write": 0, "read": 0 } }, "cost": 0, "id": "prt_0acd948dc001lAMandoxJZS2gC", "sessionID": "ses_f5326ca6affe1PlflpofN935LJ", "messageID": "msg_0acd9362a0016b7bN0bi4Mvj4w" } ] }, { "info": { "parentID": "msg_0acd935a80017b2mWSKcEoD3hb", "role": "assistant", "mode": "nudge-generation", "agent": 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} }, "id": "prt_0acdad6cf001DwA95HGCfefB8g", "sessionID": "ses_f5326ca6affe1PlflpofN935LJ", "messageID": "msg_0acda4725001KFzxqR6eGPMlmA" }, { "type": "tool", "tool": "apply_patch", "callID": "call_Sa13ZLvzcqW1yliQLE8AZ5Fa", "state": { "status": "completed", "input": { "patchText": "*** Begin Patch\n*** Add File: /var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_solution.cpp\n+#include \n+using namespace std;\n+\n+using ll = long long;\n+\n+void setIO() {\n+ ios::sync_with_stdio(false);\n+ cin.tie(nullptr);\n+}\n+\n+int main() {\n+ setIO();\n+\n+ int t;\n+ cin >> t;\n+ while (t--) {\n+ int n;\n+ cin >> n;\n+\n+ string s(n, '0');\n+ if (n <= 2) {\n+ s[0] = '1';\n+ } else {\n+ int q = n / 3;\n+ int first = (q % 2 == 1 ? q : q + 1);\n+ int second = 2 * q + 1;\n+ s[first - 1] = '1';\n+ s[second - 1] = '1';\n+ if (n % 3 == 2) {\n+ s[n - 1] = '1';\n+ }\n+ }\n+ cout << s << '\\n';\n+ }\n+}\n*** Add File: /var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_include/bits/stdc++.h\n+#include \n+#include \n*** Add File: /var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_verify.py\n+import subprocess\n+\n+\n+EXECUTABLE = \"/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_solution\"\n+\n+\n+def run(ns):\n+ assert len(ns) <= 10_000 and sum(ns) <= 200_000\n+ data = str(len(ns)) + \"\\n\" + \"\\n\".join(map(str, ns)) + \"\\n\"\n+ result = subprocess.run(\n+ [EXECUTABLE], input=data, text=True, capture_output=True, check=True\n+ )\n+ strings = result.stdout.splitlines()\n+ assert len(strings) == len(ns)\n+ for n, s in zip(ns, strings):\n+ assert len(s) == n and set(s) <= {\"0\", \"1\"}\n+ assert s.count(\"1\") <= 3\n+ counts = [1, 0, 0]\n+ residue = 0\n+ for pos, bit in enumerate(s, 1):\n+ if bit == \"1\":\n+ residue = (residue + (1 if pos % 2 else -1)) % 3\n+ counts[residue] += 1\n+ if n != 2:\n+ assert max(counts) - min(counts) <= 1, (n, counts)\n+ else:\n+ assert sum(c * (c - 1) // 2 for c in counts) == 1\n+ return strings\n+\n+\n+def count_directly(s):\n+ answer = 0\n+ for left in range(len(s)):\n+ value = 0\n+ for bit in s[left:]:\n+ value = (2 * value + int(bit)) % 3\n+ answer += value == 0\n+ return answer\n+\n+\n+small = run(list(range(1, 501)))\n+for n in range(1, 15):\n+ best = min(\n+ count_directly(format(mask, f\"0{n}b\")) for mask in range(1 << n)\n+ )\n+ assert count_directly(small[n - 1]) == best, (n, small[n - 1], best)\n+print(\"Exhaustive global-optimum checks passed for n = 1..14.\")\n+\n+for n in range(3, 200_001):\n+ q = n // 3\n+ positions = [q if q % 2 else q + 1, 2 * q + 1]\n+ if n % 3 == 2:\n+ positions.append(n)\n+ assert positions == sorted(set(positions))\n+ assert 1 <= positions[0] <= positions[-1] <= n\n+ counts = [0, 0, 0]\n+ residue = last = 0\n+ for pos in positions:\n+ counts[residue] += pos - last\n+ residue = (residue + (1 if pos % 2 else -1)) % 3\n+ last = pos\n+ counts[residue] += n + 1 - last\n+ assert sum(counts) == n + 1\n+ assert max(counts) - min(counts) <= 1, (n, counts)\n+print(\"Construction checks passed for every n = 3..200000.\")\n+\n+run([200_000])\n+run([199_999, 1])\n+run([199_998, 2])\n+run([i % 6 + 1 for i in range(10_000)])\n+print(\"Compiled solution passed maximum-length and maximum-test-count checks.\")\n*** End Patch" }, "output": "Success. Updated the following files:\nA var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_solution.cpp\nA var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_include/bits/stdc++.h\nA var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_verify.py", "metadata": { "diff": "Index: /var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_solution.cpp\n===================================================================\n--- /var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_solution.cpp\n+++ /var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_solution.cpp\n@@ -0,0 +1,35 @@\n+#include \n+using namespace std;\n+\n+using ll = long long;\n+\n+void setIO() {\n+ ios::sync_with_stdio(false);\n+ cin.tie(nullptr);\n+}\n+\n+int main() {\n+ setIO();\n+\n+ int t;\n+ cin >> t;\n+ while (t--) {\n+ int n;\n+ cin >> n;\n+\n+ string s(n, '0');\n+ if (n <= 2) {\n+ s[0] = '1';\n+ } else {\n+ int q = n / 3;\n+ int first = (q % 2 == 1 ? q : q + 1);\n+ int second = 2 * q + 1;\n+ s[first - 1] = '1';\n+ s[second - 1] = '1';\n+ if (n % 3 == 2) {\n+ s[n - 1] = '1';\n+ }\n+ }\n+ cout << s << '\\n';\n+ }\n+}\n\nIndex: /var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_include/bits/stdc++.h\n===================================================================\n--- /var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_include/bits/stdc++.h\n+++ /var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_include/bits/stdc++.h\n@@ -0,0 +1,2 @@\n+#include \n+#include \n\nIndex: /var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_verify.py\n===================================================================\n--- /var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_verify.py\n+++ /var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_verify.py\n@@ -0,0 +1,71 @@\n+import subprocess\n+\n+\n+EXECUTABLE = \"/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_solution\"\n+\n+\n+def run(ns):\n+ assert len(ns) <= 10_000 and sum(ns) <= 200_000\n+ data = str(len(ns)) + \"\\n\" + \"\\n\".join(map(str, ns)) + \"\\n\"\n+ result = subprocess.run(\n+ [EXECUTABLE], input=data, text=True, capture_output=True, check=True\n+ )\n+ strings = result.stdout.splitlines()\n+ assert len(strings) == len(ns)\n+ for n, s in zip(ns, strings):\n+ assert len(s) == n and set(s) <= {\"0\", \"1\"}\n+ assert s.count(\"1\") <= 3\n+ counts = [1, 0, 0]\n+ residue = 0\n+ for pos, bit in enumerate(s, 1):\n+ if bit == \"1\":\n+ residue = (residue + (1 if pos % 2 else -1)) % 3\n+ counts[residue] += 1\n+ if n != 2:\n+ assert max(counts) - min(counts) <= 1, (n, counts)\n+ else:\n+ assert sum(c * (c - 1) // 2 for c in counts) == 1\n+ return strings\n+\n+\n+def count_directly(s):\n+ answer = 0\n+ for left in range(len(s)):\n+ value = 0\n+ for bit in s[left:]:\n+ value = (2 * value + int(bit)) % 3\n+ answer += value == 0\n+ return answer\n+\n+\n+small = run(list(range(1, 501)))\n+for n in range(1, 15):\n+ best = min(\n+ count_directly(format(mask, f\"0{n}b\")) for mask in range(1 << n)\n+ )\n+ assert count_directly(small[n - 1]) == best, (n, small[n - 1], best)\n+print(\"Exhaustive global-optimum checks passed for n = 1..14.\")\n+\n+for n in range(3, 200_001):\n+ q = n // 3\n+ positions = [q if q % 2 else q + 1, 2 * q + 1]\n+ if n % 3 == 2:\n+ positions.append(n)\n+ assert positions == sorted(set(positions))\n+ assert 1 <= positions[0] <= positions[-1] <= n\n+ counts = [0, 0, 0]\n+ residue = last = 0\n+ for pos in positions:\n+ counts[residue] += pos - last\n+ residue = (residue + (1 if pos % 2 else -1)) % 3\n+ last = pos\n+ counts[residue] += n + 1 - last\n+ assert sum(counts) == n + 1\n+ assert max(counts) - min(counts) <= 1, (n, counts)\n+print(\"Construction checks passed for every n = 3..200000.\")\n+\n+run([200_000])\n+run([199_999, 1])\n+run([199_998, 2])\n+run([i % 6 + 1 for i in range(10_000)])\n+print(\"Compiled solution passed maximum-length and maximum-test-count checks.\")\n\n", "files": [ { "filePath": "/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_solution.cpp", "relativePath": "var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_solution.cpp", "type": "add", "patch": "Index: /var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_solution.cpp\n===================================================================\n--- /var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_solution.cpp\n+++ /var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_solution.cpp\n@@ -0,0 +1,35 @@\n+#include \n+using namespace std;\n+\n+using ll = long long;\n+\n+void setIO() {\n+ ios::sync_with_stdio(false);\n+ cin.tie(nullptr);\n+}\n+\n+int main() {\n+ setIO();\n+\n+ int t;\n+ cin >> t;\n+ while (t--) {\n+ int n;\n+ cin >> n;\n+\n+ string s(n, '0');\n+ if (n <= 2) {\n+ s[0] = '1';\n+ } else {\n+ int q = n / 3;\n+ int first = (q % 2 == 1 ? q : q + 1);\n+ int second = 2 * q + 1;\n+ s[first - 1] = '1';\n+ s[second - 1] = '1';\n+ if (n % 3 == 2) {\n+ s[n - 1] = '1';\n+ }\n+ }\n+ cout << s << '\\n';\n+ }\n+}\n", "additions": 35, "deletions": 0 }, { "filePath": "/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_include/bits/stdc++.h", "relativePath": "var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_include/bits/stdc++.h", "type": "add", "patch": "Index: /var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_include/bits/stdc++.h\n===================================================================\n--- /var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_include/bits/stdc++.h\n+++ /var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_include/bits/stdc++.h\n@@ -0,0 +1,2 @@\n+#include \n+#include \n", "additions": 2, "deletions": 0 }, { "filePath": "/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_verify.py", "relativePath": "var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_verify.py", "type": "add", "patch": "Index: /var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_verify.py\n===================================================================\n--- /var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_verify.py\n+++ /var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_verify.py\n@@ -0,0 +1,71 @@\n+import subprocess\n+\n+\n+EXECUTABLE = \"/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_solution\"\n+\n+\n+def run(ns):\n+ assert len(ns) <= 10_000 and sum(ns) <= 200_000\n+ data = str(len(ns)) + \"\\n\" + \"\\n\".join(map(str, ns)) + \"\\n\"\n+ result = subprocess.run(\n+ [EXECUTABLE], input=data, text=True, capture_output=True, check=True\n+ )\n+ strings = result.stdout.splitlines()\n+ assert len(strings) == len(ns)\n+ for n, s in zip(ns, strings):\n+ assert len(s) == n and set(s) <= {\"0\", \"1\"}\n+ assert s.count(\"1\") <= 3\n+ counts = [1, 0, 0]\n+ residue = 0\n+ for pos, bit in enumerate(s, 1):\n+ if bit == \"1\":\n+ residue = (residue + (1 if pos % 2 else -1)) % 3\n+ counts[residue] += 1\n+ if n != 2:\n+ assert max(counts) - min(counts) <= 1, (n, counts)\n+ else:\n+ assert sum(c * (c - 1) // 2 for c in counts) == 1\n+ return strings\n+\n+\n+def count_directly(s):\n+ answer = 0\n+ for left in range(len(s)):\n+ value = 0\n+ for bit in s[left:]:\n+ value = (2 * value + int(bit)) % 3\n+ answer += value == 0\n+ return answer\n+\n+\n+small = run(list(range(1, 501)))\n+for n in range(1, 15):\n+ best = min(\n+ count_directly(format(mask, f\"0{n}b\")) for mask in range(1 << n)\n+ )\n+ assert count_directly(small[n - 1]) == best, (n, small[n - 1], best)\n+print(\"Exhaustive global-optimum checks passed for n = 1..14.\")\n+\n+for n in range(3, 200_001):\n+ q = n // 3\n+ positions = [q if q % 2 else q + 1, 2 * q + 1]\n+ if n % 3 == 2:\n+ positions.append(n)\n+ assert positions == sorted(set(positions))\n+ assert 1 <= positions[0] <= positions[-1] <= n\n+ counts = [0, 0, 0]\n+ residue = last = 0\n+ for pos in positions:\n+ counts[residue] += pos - last\n+ residue = (residue + (1 if pos % 2 else -1)) % 3\n+ last = pos\n+ counts[residue] += n + 1 - last\n+ assert sum(counts) == n + 1\n+ assert max(counts) - min(counts) <= 1, (n, counts)\n+print(\"Construction checks passed for every n = 3..200000.\")\n+\n+run([200_000])\n+run([199_999, 1])\n+run([199_998, 2])\n+run([i % 6 + 1 for i in range(10_000)])\n+print(\"Compiled solution passed maximum-length and maximum-test-count checks.\")\n", "additions": 71, "deletions": 0 } ], "diagnostics": {}, "truncated": false }, "title": "Success. Updated the following files:\nA var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_solution.cpp\nA var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_include/bits/stdc++.h\nA var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_verify.py", "time": { "start": 1789606456071, "end": 1789606456076 } }, "metadata": { "openai": { "itemId": "fc_022b5c5304ba44f9016aab3a12179c87d2a5fd710fcfae3655" } }, "id": "prt_0acdad75b001gII2zJvmfX9zgD", "sessionID": "ses_f5326ca6affe1PlflpofN935LJ", "messageID": "msg_0acda4725001KFzxqR6eGPMlmA" }, { "reason": "tool-calls", "type": "step-finish", "tokens": { "total": 27672, "input": 500, "output": 1306, "reasoning": 1034, "cache": { "write": 0, "read": 24832 } }, "cost": 0, "id": "prt_0acdb6b6f0016hEmk8Ra3PTd3R", "sessionID": "ses_f5326ca6affe1PlflpofN935LJ", "messageID": "msg_0acda4725001KFzxqR6eGPMlmA" } ] }, { "info": { "parentID": "msg_0acd935a80017b2mWSKcEoD3hb", "role": "assistant", "mode": "nudge-generation", "agent": "nudge-generation", "variant": "max", "path": { "cwd": "/private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-wKvvnR", "root": "/" }, "cost": 0, "tokens": { "total": 27981, "input": 2707, "output": 186, "reasoning": 0, "cache": { "write": 0, "read": 25088 } }, "modelID": "gpt-6-astra", "providerID": "openai", "time": { "created": 1789606456179, "completed": 1789606465400 }, "finish": "tool-calls", "id": "msg_0acdb6b7300161JQY6sPaaa1pl", "sessionID": "ses_f5326ca6affe1PlflpofN935LJ" }, "parts": [ { "type": "step-start", "id": "prt_0acdb7617001xZrhDwkMtWUlVZ", "sessionID": "ses_f5326ca6affe1PlflpofN935LJ", "messageID": "msg_0acdb6b7300161JQY6sPaaa1pl" }, { "type": "tool", "tool": "bash", "callID": "call_y8WQXJiYEHTjsJTnHTmbZDoD", "state": { "status": "completed", "input": { "command": "g++ -std=c++2c -O2 -Wall -Wextra -pedantic -I \"/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_include\" \"/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_solution.cpp\" -o \"/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_solution\" && python3 \"/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_verify.py\"", "timeout": 120000 }, "output": "Exhaustive global-optimum checks passed for n = 1..14.\nConstruction checks passed for every n = 3..200000.\nCompiled solution passed maximum-length and maximum-test-count checks.\n", "metadata": { "output": "Exhaustive global-optimum checks passed for n = 1..14.\nConstruction checks passed for every n = 3..200000.\nCompiled solution passed maximum-length and maximum-test-count checks.\n", "exit": 0, "truncated": false }, "title": "g++ -std=c++2c -O2 -Wall -Wextra -pedantic -I \"/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_include\" \"/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_solution.cpp\" -o \"/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_solution\" && python3 \"/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/opencode/2264d_verify.py\"", "time": { "start": 1789606464317, "end": 1789606465398 } }, "metadata": { "openai": { "itemId": "fc_021fb38bc200e2b8016aab3a3ade3c87d2b46a988bbae504ed" } }, "id": "prt_0acdb7619001lSNYzUIxbAXlxI", "sessionID": "ses_f5326ca6affe1PlflpofN935LJ", "messageID": "msg_0acdb6b7300161JQY6sPaaa1pl" }, { "reason": "tool-calls", "type": "step-finish", "tokens": { "total": 27981, "input": 2707, "output": 186, "reasoning": 0, "cache": { "write": 0, "read": 25088 } }, "cost": 0, "id": "prt_0acdb8f77001SndH8YYARr04fG", "sessionID": "ses_f5326ca6affe1PlflpofN935LJ", "messageID": "msg_0acdb6b7300161JQY6sPaaa1pl" } ] }, { "info": { "parentID": "msg_0acd935a80017b2mWSKcEoD3hb", "role": "assistant", "mode": "nudge-generation", "agent": "nudge-generation", "variant": "max", "path": { "cwd": "/private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-wKvvnR", "root": "/" }, "cost": 0, "tokens": { "total": 28160, "input": 512, "output": 94, "reasoning": 34, "cache": { "write": 0, "read": 27520 } }, "modelID": "gpt-6-astra", "providerID": "openai", "time": { "created": 1789606465402, "completed": 1789606474441 }, "finish": "tool-calls", "id": "msg_0acdb8f7a001ZJdxfRzxW9kLTj", "sessionID": "ses_f5326ca6affe1PlflpofN935LJ" }, "parts": [ { "type": "step-start", "id": "prt_0acdb9736001LTrGda4ESrnaD3", "sessionID": "ses_f5326ca6affe1PlflpofN935LJ", "messageID": "msg_0acdb8f7a001ZJdxfRzxW9kLTj" }, { "type": "reasoning", "text": "**Preparing construction proof**", "time": { "start": 1789606467383, "end": 1789606474182 }, "metadata": { "openai": { "itemId": "rs_0037b78537817d60016aab3a4356f887d2846bdad751ca43e2", "reasoningEncryptedContent": 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\"high\"\n },\n {\n \"content\": \"Write five progressive hints, a self-contained editorial, and the C++26 solution.\",\n \"status\": \"completed\",\n \"priority\": \"high\"\n }\n]", "metadata": { "todos": [ { "content": "Read the supplied tutorial and confirm the approach matches the statement.", "status": "completed", "priority": "high" }, { "content": "Simplify the construction and verify optimality and edge cases.", "status": "completed", "priority": "high" }, { "content": "Write five progressive hints, a self-contained editorial, and the C++26 solution.", "status": "completed", "priority": "high" } ], "truncated": false }, "title": "0 todos", "time": { "start": 1789606571228, "end": 1789606571233 } }, "metadata": { "openai": { "itemId": "fc_0fd1f4aa662c6864016aab3aa9f5a887d1b6adf67864a16e4c" } }, "id": "prt_0acdd2866001U9394SL71aW3Ci", "sessionID": "ses_f5326ca6affe1PlflpofN935LJ", "messageID": "msg_0acdbb2cc0015Uc7boNlDG6DCj" }, { "reason": "tool-calls", "type": "step-finish", "tokens": { "total": 31451, "input": 510, "output": 93, "reasoning": 3072, "cache": { "write": 0, "read": 27776 } }, "cost": 0, "id": "prt_0acdd2d5e001d32PzTctfuWD6z", "sessionID": "ses_f5326ca6affe1PlflpofN935LJ", "messageID": "msg_0acdbb2cc0015Uc7boNlDG6DCj" } ] }, { "info": { "parentID": "msg_0acd935a80017b2mWSKcEoD3hb", "role": "assistant", "mode": "nudge-generation", "agent": "nudge-generation", "variant": "max", "path": { "cwd": "/private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-wKvvnR", "root": "/" }, "cost": 0, "tokens": { "total": 33894, "input": 3544, "output": 2318, "reasoning": 0, "cache": { "write": 0, "read": 28032 } }, "modelID": "gpt-6-astra", "providerID": "openai", "time": { "created": 1789606571376, "completed": 1789606644690 }, "finish": "tool-calls", "structured": { "status": "success", "reason": null, "hints": [ { "order": 1, "content": "You don't need the binary integers themselves—only their remainders modulo $3$. What pattern do $1,2,4,8,\\ldots$ follow modulo $3$?" }, { "order": 2, "content": "Since $2\\equiv-1\\pmod3$, try alternating prefix sums:\n$$p_0=0,\\qquad p_i\\equiv\\sum_{j=1}^{i}(-1)^{j-1}s_j\\pmod3.$$\nShow that substring $[l,r]$ is irritating exactly when $p_{l-1}=p_r$. Don't forget the empty prefix $p_0$." }, { "order": 3, "content": "Let $c_0,c_1,c_2$ count the occurrences of each remainder among the $n+1$ prefixes. Then\n$$f(s)=\\binom{c_0}{2}+\\binom{c_1}{2}+\\binom{c_2}{2}.$$\nFor a fixed total, this expression is minimized when the counts differ by at most one. That gives a lower bound for **all** binary strings, regardless of how many ones they contain." }, { "order": 4, "content": "A zero leaves the alternating prefix remainder unchanged. A one adds $1$ at an odd position and subtracts $1$ at an even position. Thus, two ones at odd positions create three constant blocks of prefix remainders: $0$, then $1$, then $2$. Aim for nearly equal block lengths, using a third one at the end to fix the imbalance. It helps to first construct a length $3q+2$ string whose three counts are exactly equal." }, { "order": 5, "content": "For $n\\ge3$, let $q=\\lfloor n/3\\rfloor$ and imagine a string of length $N=3q+2$. Put ones at these **1-based** positions:\n- $a=q$ if $q$ is odd, otherwise $a=q+1$;\n- $b=2q+1$;\n- $N$.\nEach prefix remainder occurs exactly $q+1$ times. Keep only the first $n$ characters. This removes at most two prefix remainders, and the last two have different values, so the counts stay balanced. Handle $n=1,2$ with `1` and `10`." } ], "editorial": "Zeroes aren't free here: every all-zero substring still annoys the doctor. The trick is to turn the substring count into a problem about balancing three numbers.\n\n### 1. Divisible substrings are pairs of equal prefixes\n\nUse 1-based positions. Because\n$$2\\equiv-1\\pmod3,$$\nthe powers of two alternate between $1$ and $-1$ modulo $3$.\n\nDefine alternating prefix remainders, always represented by $0$, $1$, or $2$:\n$$p_0=0,\\qquad p_i\\equiv\\sum_{j=1}^{i}(-1)^{j-1}s_j\\pmod3.$$\n\nFor the binary value $V(l,r)$ of substring $[l,r]$,\n$$\n\\begin{aligned}\nV(l,r)\n&\\equiv\\sum_{j=l}^{r}s_j(-1)^{r-j}\\\\\n&\\equiv(-1)^{r-1}(p_r-p_{l-1})\\pmod3.\n\\end{aligned}\n$$\nMultiplying by $1$ or $-1$ does not change whether a value is zero modulo $3$. Therefore,\n$$V(l,r)\\equiv0\\pmod3\\iff p_{l-1}=p_r.$$\n\nLet $c_0,c_1,c_2$ be the counts of the three remainders among **all $n+1$ prefixes**, including $p_0$. Each pair of equal prefixes at indices $i0.\n$$\nRepeating this balancing step gives three counts differing by at most one.\n\nThis is a lower bound for **every binary string**, including strings with more than three ones. If our sparse construction reaches it, we've proved global optimality—not merely optimality among sparse strings.\n\n### 3. The construction\n\nFor $n=1$, output `1`. For $n=2$, output `10`.\n\nFor $n\\ge3$, let\n$$q=\\left\\lfloor\\frac n3\\right\\rfloor.$$\nStart with $n$ zeroes and put ones at the following 1-based positions:\n$$\na=\\begin{cases}\nq,&q\\text{ is odd},\\\\\nq+1,&q\\text{ is even},\n\\end{cases}\n\\qquad b=2q+1.\n$$\nIf $n\\bmod3=2$, also put a one at position $n$.\n\nThat's the whole algorithm. No search or substring counting is needed in the implementation. Convert the positions to zero-based indices when accessing the C++ string.\n\n### 4. Why these positions work\n\nA zero keeps the prefix remainder unchanged. A one changes it by $+1$ at an odd position and by $-1$ at an even position.\n\nTo understand the construction, first imagine a slightly longer string of length\n$$N=3q+2$$\nwith ones at $a$, $b$, and $N$. The desired length $n$ is obtained by deleting zero, one, or two characters from its end.\n\nBoth $a$ and $b$ are odd. Consequently, before the final character, the prefix remainders form these blocks:\n\n- $p_0,\\ldots,p_{a-1}$ are $0$: there are $a$ of them.\n- $p_a,\\ldots,p_{b-1}$ are $1$: there are $b-a$ of them.\n- $p_b,\\ldots,p_{N-1}$ are $2$: there are $N-b=q+1$ of them.\n\nNow consider the parity of $q$.\n\n**If $q$ is odd:** $a=q$, so the three block lengths are $q,q+1,q+1$. Also, $N$ is odd, so the final one changes remainder $2$ to $0$. It adds the missing occurrence of remainder $0$.\n\n**If $q$ is even:** $a=q+1$, so the block lengths are $q+1,q,q+1$. Now $N$ is even, so the final one changes remainder $2$ to $1$. It adds the missing occurrence of remainder $1$.\n\nIn both cases, among $p_0,\\ldots,p_N$, every remainder occurs exactly $q+1$ times.\n\n#### Trimming preserves balance\n\nKeeping only the first $n$ characters leaves all retained prefix remainders unchanged. It removes at most the final two prefixes.\n\nThose last two remainders are different:\n$$p_{N-1}=2,\\qquad p_N\\in\\{0,1\\}.$$\nThus, each remainder loses at most one occurrence. The remaining counts are all $q$ or $q+1$, so they differ by at most one and attain the global lower bound.\n\nThe first two ones always fit because $q\\ge1$ and\n$$a\nusing namespace std;\n\nusing ll = long long;\n\nvoid setIO() {\n ios::sync_with_stdio(false);\n cin.tie(nullptr);\n}\n\nint main() {\n setIO();\n\n int t;\n cin >> t;\n while (t--) {\n int n;\n cin >> n;\n\n string s(n, '0');\n if (n <= 2) {\n s[0] = '1';\n } else {\n int q = n / 3;\n int first = (q % 2 == 1 ? q : q + 1);\n int second = 2 * q + 1;\n s[first - 1] = '1';\n s[second - 1] = '1';\n if (n % 3 == 2) {\n s[n - 1] = '1';\n }\n }\n cout << s << '\\n';\n }\n}" }, "id": "msg_0acdd2d70001B76J7a18OgCy9L", "sessionID": "ses_f5326ca6affe1PlflpofN935LJ" }, "parts": [ { "type": "step-start", "id": "prt_0acdd3cc8001YS6hsiW3yrmuIf", "sessionID": "ses_f5326ca6affe1PlflpofN935LJ", "messageID": "msg_0acdd2d70001B76J7a18OgCy9L" }, { "type": "tool", "tool": "StructuredOutput", "callID": "call_pKwtRS3yRDVmuLx5BkqgBKoy", "state": { "status": "completed", "input": { "status": "success", "reason": null, "hints": [ { "order": 1, "content": "You don't need the binary integers themselves—only their remainders modulo $3$. What pattern do $1,2,4,8,\\ldots$ follow modulo $3$?" }, { "order": 2, "content": "Since $2\\equiv-1\\pmod3$, try alternating prefix sums:\n$$p_0=0,\\qquad p_i\\equiv\\sum_{j=1}^{i}(-1)^{j-1}s_j\\pmod3.$$\nShow that substring $[l,r]$ is irritating exactly when $p_{l-1}=p_r$. Don't forget the empty prefix $p_0$." }, { "order": 3, "content": "Let $c_0,c_1,c_2$ count the occurrences of each remainder among the $n+1$ prefixes. Then\n$$f(s)=\\binom{c_0}{2}+\\binom{c_1}{2}+\\binom{c_2}{2}.$$\nFor a fixed total, this expression is minimized when the counts differ by at most one. That gives a lower bound for **all** binary strings, regardless of how many ones they contain." }, { "order": 4, "content": "A zero leaves the alternating prefix remainder unchanged. A one adds $1$ at an odd position and subtracts $1$ at an even position. Thus, two ones at odd positions create three constant blocks of prefix remainders: $0$, then $1$, then $2$. Aim for nearly equal block lengths, using a third one at the end to fix the imbalance. It helps to first construct a length $3q+2$ string whose three counts are exactly equal." }, { "order": 5, "content": "For $n\\ge3$, let $q=\\lfloor n/3\\rfloor$ and imagine a string of length $N=3q+2$. Put ones at these **1-based** positions:\n- $a=q$ if $q$ is odd, otherwise $a=q+1$;\n- $b=2q+1$;\n- $N$.\nEach prefix remainder occurs exactly $q+1$ times. Keep only the first $n$ characters. This removes at most two prefix remainders, and the last two have different values, so the counts stay balanced. Handle $n=1,2$ with `1` and `10`." } ], "editorial": "Zeroes aren't free here: every all-zero substring still annoys the doctor. The trick is to turn the substring count into a problem about balancing three numbers.\n\n### 1. Divisible substrings are pairs of equal prefixes\n\nUse 1-based positions. Because\n$$2\\equiv-1\\pmod3,$$\nthe powers of two alternate between $1$ and $-1$ modulo $3$.\n\nDefine alternating prefix remainders, always represented by $0$, $1$, or $2$:\n$$p_0=0,\\qquad p_i\\equiv\\sum_{j=1}^{i}(-1)^{j-1}s_j\\pmod3.$$\n\nFor the binary value $V(l,r)$ of substring $[l,r]$,\n$$\n\\begin{aligned}\nV(l,r)\n&\\equiv\\sum_{j=l}^{r}s_j(-1)^{r-j}\\\\\n&\\equiv(-1)^{r-1}(p_r-p_{l-1})\\pmod3.\n\\end{aligned}\n$$\nMultiplying by $1$ or $-1$ does not change whether a value is zero modulo $3$. Therefore,\n$$V(l,r)\\equiv0\\pmod3\\iff p_{l-1}=p_r.$$\n\nLet $c_0,c_1,c_2$ be the counts of the three remainders among **all $n+1$ prefixes**, including $p_0$. Each pair of equal prefixes at indices $i0.\n$$\nRepeating this balancing step gives three counts differing by at most one.\n\nThis is a lower bound for **every binary string**, including strings with more than three ones. If our sparse construction reaches it, we've proved global optimality—not merely optimality among sparse strings.\n\n### 3. The construction\n\nFor $n=1$, output `1`. For $n=2$, output `10`.\n\nFor $n\\ge3$, let\n$$q=\\left\\lfloor\\frac n3\\right\\rfloor.$$\nStart with $n$ zeroes and put ones at the following 1-based positions:\n$$\na=\\begin{cases}\nq,&q\\text{ is odd},\\\\\nq+1,&q\\text{ is even},\n\\end{cases}\n\\qquad b=2q+1.\n$$\nIf $n\\bmod3=2$, also put a one at position $n$.\n\nThat's the whole algorithm. No search or substring counting is needed in the implementation. Convert the positions to zero-based indices when accessing the C++ string.\n\n### 4. Why these positions work\n\nA zero keeps the prefix remainder unchanged. A one changes it by $+1$ at an odd position and by $-1$ at an even position.\n\nTo understand the construction, first imagine a slightly longer string of length\n$$N=3q+2$$\nwith ones at $a$, $b$, and $N$. The desired length $n$ is obtained by deleting zero, one, or two characters from its end.\n\nBoth $a$ and $b$ are odd. Consequently, before the final character, the prefix remainders form these blocks:\n\n- $p_0,\\ldots,p_{a-1}$ are $0$: there are $a$ of them.\n- $p_a,\\ldots,p_{b-1}$ are $1$: there are $b-a$ of them.\n- $p_b,\\ldots,p_{N-1}$ are $2$: there are $N-b=q+1$ of them.\n\nNow consider the parity of $q$.\n\n**If $q$ is odd:** $a=q$, so the three block lengths are $q,q+1,q+1$. Also, $N$ is odd, so the final one changes remainder $2$ to $0$. It adds the missing occurrence of remainder $0$.\n\n**If $q$ is even:** $a=q+1$, so the block lengths are $q+1,q,q+1$. Now $N$ is even, so the final one changes remainder $2$ to $1$. It adds the missing occurrence of remainder $1$.\n\nIn both cases, among $p_0,\\ldots,p_N$, every remainder occurs exactly $q+1$ times.\n\n#### Trimming preserves balance\n\nKeeping only the first $n$ characters leaves all retained prefix remainders unchanged. It removes at most the final two prefixes.\n\nThose last two remainders are different:\n$$p_{N-1}=2,\\qquad p_N\\in\\{0,1\\}.$$\nThus, each remainder loses at most one occurrence. The remaining counts are all $q$ or $q+1$, so they differ by at most one and attain the global lower bound.\n\nThe first two ones always fit because $q\\ge1$ and\n$$a\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, '0');\n if (n <= 2) {\n s[0] = '1';\n } else {\n int q = n / 3;\n int first = (q % 2 == 1 ? q : q + 1);\n int second = 2 * q + 1;\n s[first - 1] = '1';\n s[second - 1] = '1';\n if (n % 3 == 2) {\n s[n - 1] = '1';\n }\n }\n cout << s << '\\n';\n }\n}" }, "output": "Structured output captured successfully.", "metadata": { "valid": true }, "title": "Structured Output", "time": { "start": 1789606644679, "end": 1789606644681 } }, "metadata": { "openai": { "itemId": "fc_0b7bb28384c32a89016aab3aaf1fd487d1bf41b8143daeb685" } }, "id": "prt_0acdd3ccc001WZB8no3yHl9zSf", "sessionID": "ses_f5326ca6affe1PlflpofN935LJ", "messageID": "msg_0acdd2d70001B76J7a18OgCy9L" }, { "reason": "tool-calls", "type": "step-finish", "tokens": { "total": 33894, "input": 3544, "output": 2318, "reasoning": 0, "cache": { "write": 0, "read": 28032 } }, "cost": 0, "id": "prt_0acde4bcf001B7JzsGjhb7hNwX", "sessionID": "ses_f5326ca6affe1PlflpofN935LJ", "messageID": "msg_0acdd2d70001B76J7a18OgCy9L" } ] } ] }