{ "info": { "id": "ses_f4dc5c871ffeijw4TBX8KF336h", "slug": "neon-pixel", "projectID": "global", "directory": "/private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-XtZbAE", "path": "private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-XtZbAE", "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": 23637, "output": 1661, "reasoning": 42, "cache": { "read": 31488, "write": 0 } }, "time": { "created": 1789696554894, "updated": 1789696625071 } }, "messages": [ { "info": { "role": "user", "time": { "created": 1789696554913 }, "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_0b23a37a1001LUqDes84ANGaSC", "sessionID": "ses_f4dc5c871ffeijw4TBX8KF336h" }, "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 2254B: \"Evanescent\" (rated 900).\n\nKnown problem metadata:\n- Contest ID: 2254\n- Index: B\n- Name: Evanescent\n- Rating: 900 (Codeforces difficulty rating)\n- Tags: implementation, strings, two pointers\n- Problem URL: https://codeforces.com/contest/2254/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. Evanescent
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

Let $$$f(s)$$$ be the compressed version of a string $$$s$$$, formed by replacing every maximal contiguous block of identical characters with a single copy of that character. For example, $$$f($$$\"aabbcc\"$$$) \\ = \\ $$$\"abc\".

Let $$$|s|$$$ denote the length of a string $$$s$$$. Following this, $$$|f(s)|$$$ denotes the length of the compressed string. For example:

Yousef has given you a string $$$s$$$ consisting of $$$n$$$ lowercase Latin letters. You must delete exactly one character $$$s_i$$$ ($$$2 \\le i \\le n - 1$$$) to form a new string $$$s'$$$, and then find the minimum possible value of $$$|f(s')|$$$.

Note that you cannot delete $$$s_1$$$ or $$$s_n$$$.

Input

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

The first line of each test case contains an integer $$$n$$$ ($$$3 \\le n \\le 2 \\cdot 10^5$$$) — the length of the string.

The second line of each test case contains a string $$$s$$$ ($$$|s| = n$$$), consisting of lowercase Latin letters.

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 — the minimum possible length of the resulting compressed string after deleting one character.

Example
Input
9
3
abb
3
aab
3
abc
4
abaa
4
abba
5
eeeee
6
yyssee
7
abacaba
18
goodluckandhavefun
Output
2
2
2
1
3
1
3
5
16
Note

In the first test case, we can only delete the character $$$s_2 = $$$ 'b', producing a string $$$s' = $$$ \"ab\", with $$$|f(s')| = 2$$$. Therefore, $$$2$$$ is the minimum length achievable.

In the fourth test case, we can delete the character $$$s_2 = $$$ 'b'. The resulting string is $$$s' = $$$ \"aaa\" with $$$f(s') = $$$ \"a\", so $$$|f(s')| = 1$$$.

In the sixth test case, deleting any valid character results in $$$f(s') = $$$ \"e\" and $$$|f(s')| = 1$$$.

In the eighth test case, we can delete the character $$$s_4 = $$$ 'c'. The resulting string is $$$s' = $$$ \"abaaba\" with $$$f(s') = $$$ \"ababa\" and $$$|f(s')| = 5$$$.

\n\n\n\nSource lookup status:\n\n- Tutorial (en): https://codeforces.com/blog/entry/155666 loaded (200 OK; title: \"Codeforces Round 1114 (Div. 3) — Editorial - Codeforces\"; does not mention 2254B)\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_0b23a37a2001zfgmI62wff29MS", "sessionID": "ses_f4dc5c871ffeijw4TBX8KF336h", "messageID": "msg_0b23a37a1001LUqDes84ANGaSC" } ] }, { "info": { "parentID": "msg_0b23a37a1001LUqDes84ANGaSC", "role": "assistant", "mode": "nudge-generation", "agent": "nudge-generation", "variant": "low", "path": { "cwd": "/private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-XtZbAE", "root": "/" }, "cost": 0, "tokens": { "total": 9275, "input": 9243, "output": 32, "reasoning": 0, "cache": { "write": 0, "read": 0 } }, "modelID": "gpt-6-astra", "providerID": "openai", "time": { "created": 1789696555045, "completed": 1789696567428 }, "finish": "tool-calls", "id": "msg_0b23a3825001URCaZVf4A1Iyie", "sessionID": "ses_f4dc5c871ffeijw4TBX8KF336h" }, "parts": [ { "type": "step-start", "id": "prt_0b23a407c001zFploTDiqI9Wql", "sessionID": "ses_f4dc5c871ffeijw4TBX8KF336h", "messageID": "msg_0b23a3825001URCaZVf4A1Iyie" }, { "type": "tool", "tool": "webfetch", "callID": "call_r6odM4lRF3PajcMhtJ88YCoR", "state": { "status": "completed", "input": { "url": "https://codeforces.com/blog/entry/155666", "format": "markdown" }, "output": "---\nimage: https://codeforces.com/codeforces.org/s/66324/images/codeforces-telegram-square-1024x1024.png\ntitle: Codeforces Round 1114 (Div. 3) — Editorial - Codeforces\ndescription: Codeforces. Programming competitions and contests, programming community\n---\n\n \n\n \n[![Codeforces](//codeforces.com/codeforces.org/s/66324/images/codeforces-logo-with-telegram-555x130.png \"Codeforces\")](https://codeforces.com/) \n\n[![In English](//codeforces.com/codeforces.org/s/66324/images/flags/24/gb.png \"In English\")](https://codeforces.com/blog/entry/155666?locale=en) [![По-русски](//codeforces.com/codeforces.org/s/66324/images/flags/24/ru.png \"По-русски\")](https://codeforces.com/blog/entry/155666?locale=ru) \n\n[Enter](https://codeforces.com/enter?back=%2Fblog%2Fentry%2F155666)| [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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![Necropost](//codeforces.com/codeforces.org/s/66324/images/icons/hourglass.png \"Necropost\")\n\n| | [Detailed →](https://codeforces.com/recent-actions) |\n| | --------------------------------------------------- |\n\n* [yse](https://codeforces.com/profile/yse)\n* [Blog](https://codeforces.com/blog/yse)\n* [Teams](https://codeforces.com/teams/with/yse)\n* [Submissions](https://codeforces.com/submissions/yse)\n* [Groups](https://codeforces.com/groups/with/yse)\n* [Contests](https://codeforces.com/contests/with/yse)\n* [Problemsetting](https://codeforces.com/contests/writer/yse)\n\n### [yse's blog](https://codeforces.com/blog/yse)\n\n[ Codeforces Round 1114 (Div. 3) — Editorial ](https://codeforces.com/blog/entry/155666) \n\nBy [yse](https://codeforces.com/profile/yse \"Candidate Master yse\"), 7 weeks ago, ![In English](//codeforces.com/codeforces.org/s/66324/images/flags/24/gb.png \"In English\") \n\nThank you for participating in the contest! I hope you enjoyed all problems :) Also, thanks to [reirugan](https://codeforces.com/profile/reirugan \"Candidate Master reirugan\") for proofreading the editorial.\n\n**Rating Predictions**\n\n| Predictor | A | B | C1 C2 | D | E | F | G |\n| --------------------------------------------------------------------------------------------- | --- | ---- | ------------- | ---- | ---- | ---- | ---- |\n| [yse](https://codeforces.com/profile/yse \"Candidate Master yse\") | 800 | 800 | (900 — 1000) | 1200 | 1400 | 1700 | 2000 |\n| [Wageeh](https://codeforces.com/profile/Wageeh \"Expert Wageeh\") | — | 1000 | (1300 — 1300) | 1500 | — | — | 1900 |\n| [Proof\\_by\\_QED](https://codeforces.com/profile/Proof%5Fby%5FQED \"Master Proof_by_QED\") | 800 | 800 | (1000 — 1100) | 1300 | 1300 | 1900 | 2100 |\n| [detective...dots](https://codeforces.com/profile/detective...dots \"Expert detective...dots\") | 800 | 900 | (1100 — 1100) | 1300 | 1400 | 1800 | 2100 |\n| [Mr\\_Bald](https://codeforces.com/profile/Mr%5FBald \"Specialist Mr_Bald\") | 800 | 900 | (1100 — 1200) | 1400 | 1500 | 1800 | — |\n| [Argentum47](https://codeforces.com/profile/Argentum47 \"Candidate Master Argentum47\") | 800 | 900 | (1000 — 1200) | 1300 | 1400 | 1800 | 2000 |\n\n**Rate the contest!**\n\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Excellent contest\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Good contest\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Average contest\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Bad contest\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Horrible contest\n\n**Rate the difficulty!**\n\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Easy contest\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Normal contest\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Hard contest\n\n#### [A — Riptide](https://codeforces.com/contest/2254/problem/A)\n\n**Hint**\n\nAssume $$$a \\\\le b \\\\le c$$$, does $$$b$$$ ever change?\n\n**Solution**\n\nFor simplicity, let's assume $$$a \\\\le b \\\\le c$$$ (we can sort the input to achieve this). The operation decreases $$$c$$$ by $$$1$$$ and increases $$$a$$$ by $$$1$$$. Therefore, $$$a$$$ is getting closer to $$$b$$$ and $$$c$$$ is also getting closer to $$$b$$$. The round ends when $$$2$$$ players have the same value, so the answer is $$$\\\\min(b - a, c - b)$$$ since if either $$$a$$$ or $$$c$$$ become equal to $$$b$$$, the game will end.\n\n**Code (C++)**\n\n```\n#include \nusing namespace std;\n\nint main() {\n int t;\n cin >> t;\n while(t--) {\n vector v(3);\n for(int &x : v) cin >> x;\n sort(v.begin(), v.end());\n cout << min(v[2] - v[1], v[1] - v[0]) << endl;\n }\n}\n\n```\n\n**Rate the problem!**\n\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Excellent problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Good problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Average problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Bad problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Horrible problem\n\n#### [B — Evanescent](https://codeforces.com/contest/2254/problem/B)\n\n**Hint**\n\nIf we perform the operation on any character in a maximal contiguous block of length $$$ \\\\gt 1$$$, then $$$f(s') = f(s)$$$.\n\n**Solution**\n\nThe function $$$f(s)$$$ converts every maximal contiguous block into a block of size $$$1$$$. Therefore, removing a character from a block does not change anything if the block will still exist. The only way we can delete the whole block is if it had a size of $$$1$$$. This allows to delete the sole character that exists in the block, additionally achieving $$$f(s') \\\\lt f(s)$$$.\n\nWhen deleting a block of size $$$1$$$, we should consider what happens when the left block touches the right block. If they both contain the same character, then they would merge into one block. Therefore, the answer would decrease by $$$2$$$ (our $$$1$$$-sized block got removed, and $$$2$$$ separate blocks got merged into one).\n\nOtherwise, if the blocks do not contain the same character, then the answer would decrease by $$$1$$$ since nothing will happen other than removing the $$$1$$$-sized block.\n\n**Code (C++)**\n\n```\n#include \nusing namespace std;\n\nint main() {\n int t;\n cin >> t;\n while(t--) {\n int n;\n cin >> n;\n string s;\n cin >> s;\n \n int ans = 1, x = 0;\n for(int i = 1; i < n; i++) {\n if(s[i] != s[i - 1]) ans++;\n if(i == n - 1) break;\n \n if(s[i] != s[i - 1] && s[i] != s[i + 1]) {\n if(s[i + 1] == s[i - 1]) x = 2;\n else x = max(x, 1);\n }\n }\n \n cout << ans - x << endl;\n }\n}\n\n```\n\n**Rate the problem!**\n\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Excellent problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Good problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Average problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Bad problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Horrible problem\n\n#### [C1 — Marenol (easy version)](https://codeforces.com/contest/2254/problem/C1)\n\n**Hint 1**\n\nThe operation is equivalent to freely swapping $$$a\\_i$$$ with $$$a\\_{i+2}$$$ for any valid index $$$i$$$.\n\n**Hint 2**\n\nAfter the reduction from Hint 1, we can treat $$$a$$$ as two separate strings.\n\n**Solution**\n\nNotice that the operation can be treated as swapping $$$a\\_i$$$ with $$$a\\_{i+2}$$$, this works since the operation does not change the middle element, and only changes the first one and the third one.\n\nBecause an element can only move in steps of two, it can never change the parity of its index. Any character at an odd position will always remain at an odd position, and the same applies for even indices.\n\nSince we have the power to swap the characters at odd indices endlessly (as many times as we want), we can rearrange them into any arrangement we want, independently of the characters at even indices. Therefore, the elements at odd positions in $$$a$$$ can match the elements at odd positions in $$$b$$$ if they have the same number of ones (or zeros). The same can be applied for even positions.\n\n**Code (C++)**\n\n```\n#include \nusing namespace std;\n \nint main() {\n int t;\n cin >> t;\n while(t--) {\n int n;\n cin >> n;\n string a, b;\n cin >> a >> b;\n \n int cnta[2] = {}, cntb[2] = {};\n for(int i = 0; i < n; i++) {\n cnta[i % 2] += a[i] == '1';\n cntb[i % 2] += b[i] == '1';\n }\n \n cout << (cnta[0] == cntb[0] && cnta[1] == cntb[1] ? \"YES\" : \"NO\") << endl;\n }\n}\n\n```\n\n**Rate the problem!**\n\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Excellent problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Good problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Average problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Bad problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Horrible problem\n\n#### [C2 — Marenol (hard version)](https://codeforces.com/contest/2254/problem/C2)\n\n**Solution**\n\nBuilding off the core observation from the easy version, we know that elements are strictly trapped in their own parity. Odd-indexed characters only interact with odd-indexed characters, and even with even.\n\nThe problem then boils down to a classic task: what is the minimum number of adjacent swaps needed to transform a binary array into another?\n\nSince all '$$$\\\\texttt{1}$$$'s are identical, they never need to leapfrog over each other. The optimal path is always for the $$$k$$$-th '$$$\\\\texttt{1}$$$' in your starting array to walk directly to the position of the $$$k$$$-th '$$$\\\\texttt{1}$$$' in your target array.\n\nTo implement this, just iterate through both strings and collect the positions of the '$$$\\\\texttt{1}$$$'s in the odd indices into one list, and the positions of the '$$$\\\\texttt{1}$$$'s in the even indices into another. If the lengths of the corresponding lists for $$$a$$$ and $$$b$$$ don't match, it's impossible to transform, so output $$$-1$$$. Otherwise, just sum up the absolute differences between the $$$k$$$-th elements of the starting and target lists for both the odd and even arrays. That sum, divided by $$$2$$$, is your exact minimum number of operations. We divide by $$$2$$$ because one operation moves characters by $$$2$$$ steps.\n\n**Code (C++)**\n\n```\n#include \nusing namespace std;\n#define ll long long\n \nint main() {\n int t;\n cin >> t;\n while(t--) {\n int n;\n cin >> n;\n string a, b;\n cin >> a >> b;\n \n ll ans = 0;\n bool ok = true;\n \n for(int k = 0; k < 2; k++) {\n vector p1, p2;\n for(int i = k; i < n; i += 2) {\n if(a[i] == '1') p1.push_back(i);\n if(b[i] == '1') p2.push_back(i);\n }\n \n if(p1.size() != p2.size()) {\n ok = false;\n break;\n }\n \n for(int i = 0; i < p1.size(); i++) ans += abs(p1[i] - p2[i]);\n }\n \n if(!ok) cout << -1 << endl;\n else cout << ans / 2 << endl;\n }\n}\n\n```\n\n**Rate the problem!**\n\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Excellent problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Good problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Average problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Bad problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Horrible problem\n\n#### [D — Silhouette](https://codeforces.com/contest/2254/problem/D)\n\n**Hint 1**\n\nEqual values in $$$a$$$ will always have the exact same shadow, and larger elements always have strictly larger shadows. So if you sort the distinct values of $$$b$$$, they correspond in order to the distinct values of $$$a$$$: the group with the $$$i$$$-th smallest value has shadow equal to the $$$i$$$-th smallest distinct value in $$$b$$$.\n\n**Hint 2**\n\nLet $$$s\\_1 \\\\lt s\\_2 \\\\lt \\\\dots \\\\lt s\\_m$$$ be the distinct sorted values of $$$b$$$, and let group $$$i$$$ be all elements sharing the value corresponding to $$$s\\_i$$$. Then $$$s\\_i$$$ is the sum of all groups smaller than group $$$i$$$, and $$$s\\_{i+1}$$$ is that same sum plus group $$$i$$$ itself. So $$$s\\_{i+1} - s\\_i = c\\_i \\\\cdot v\\_i$$$, where $$$c\\_i$$$ is the count of elements in group $$$i$$$ and $$$v\\_i$$$ is their value.\n\n**Hint 3**\n\nWhat must $$$s\\_1$$$ be? And once every $$$v\\_i$$$ is pinned down by the equation above, is there a group left with no equation constraining its value? What's the smallest legal choice for it?\n\n**Solution**\n\nIt's easy to see that equal values in $$$a$$$ produce equal shadows. The converse also holds: distinct values in $$$a$$$ always produce distinct shadows, and moreover a larger value always produces a larger shadow. So if we sort the distinct values of $$$a$$$ as $$$v\\_1 \\\\lt v\\_2 \\\\lt \\\\dots \\\\lt v\\_k$$$ with frequencies $$$c\\_1, \\\\dots, c\\_k$$$, then the shadow of group $$$i$$$ is simply the sum of all smaller groups, $$$S\\_{i-1} = \\\\sum\\_{j \\\\lt i} c\\_j v\\_j$$$, and these prefix sums satisfy $$$S\\_0 = 0 \\\\lt S\\_1 \\\\lt \\\\dots \\\\lt S\\_{k-1}$$$. In other words, the distinct values of $$$b$$$, sorted, correspond exactly and in order to these prefix sums.\n\nThis gives a direct reconstruction. Sort the distinct values of $$$b$$$ as $$$s\\_1 \\\\lt s\\_2 \\\\lt \\\\dots \\\\lt s\\_m$$$; since $$$S\\_0$$$ must be $$$0$$$, we need $$$s\\_1 = 0$$$, otherwise the answer is $$$-1$$$. The count of $$$s\\_i$$$ in $$$b$$$ gives $$$c\\_i$$$, the size of group $$$i$$$, and since $$$s\\_{i+1} - s\\_i = c\\_i \\\\cdot v\\_i$$$, we can recover $$$v\\_i = \\\\frac{(s\\_{i+1} - s\\_i)}{c\\_i}$$$. If this division isn't exact, or if the resulting $$$v\\_i$$$ doesn't exceed $$$v\\_{i-1}$$$, no valid array exists. The very last group has no such equation to pin it down, so to keep $$$a$$$ lexicographically smallest we assign it the smallest legal value, one more than the previous group's value.\n\nOnce every group's value is known, replace each $$$b\\_i$$$ by its group's value to get $$$a$$$.\n\n**Code (C++)**\n\n```\n#include \nusing namespace std;\n#define ll long long\n \nint main() {\n int t;\n cin >> t;\n while(t--) {\n int n;\n cin >> n;\n vector b(n);\n map freq;\n \n for(int i = 0; i < n; i++) {\n cin >> b[i];\n freq[b[i]]++;\n }\n \n if (!freq.count(0)) {\n cout << -1 << endl;\n continue;\n }\n \n map ans;\n ll prv_shadow = 0, prv_cnt = 0, lst = 0;\n bool ok = true;\n \n for(auto [cur_shadow, cur_cnt] : freq) {\n if(cur_shadow != 0) {\n ll diff = cur_shadow - prv_shadow;\n \n if(diff % prv_cnt != 0) {\n ok = false;\n break;\n }\n \n ll u = diff / prv_cnt;\n if(u <= lst) {\n ok = false;\n break;\n }\n \n ans[prv_shadow] = u;\n lst = u;\n }\n \n prv_shadow = cur_shadow;\n prv_cnt = cur_cnt;\n }\n \n if(!ok) {\n cout << -1 << endl;\n continue;\n }\n \n ans[prv_shadow] = lst + 1;\n for(int i = 0; i < n; ++i) cout << ans[b[i]] << \" \\n\"[i == n - 1];\n }\n}\n\n```\n\n**Rate the problem!**\n\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Excellent problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Good problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Average problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Bad problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Horrible problem\n\n#### [E — Chronostasis](https://codeforces.com/contest/2254/problem/E)\n\n**Hint 1**\n\nThe problem states that $$$b\\_i = a\\_i - a\\_{i-1}$$$. This directly implies that $$$a\\_i = a\\_{i-1} + b\\_i$$$. If you assume $$$a\\_0 = 0$$$, you can view $$$a$$$ simply as the prefix sums of some permutation of $$$b$$$.\n\n**Hint 2**\n\nThe standard way to achieve a lexicographically smallest sequence is to build it greedily from left to right, making $$$a\\_1$$$ as small as possible, then $$$a\\_2$$$ as small as possible, and so forth.\n\n**Hint 3**\n\nSince $$$a\\_{i-1}$$$ is fixed from the previous steps, minimizing $$$a\\_i$$$ means you must pick the smallest available $$$b\\_i$$$. However, the problem imposes a strict rule: every element in $$$a$$$ must be strictly positive ($$$a\\_i \\\\ge 1$$$). How does this restrict your choice of $$$b\\_i$$$?\n\n**Solution**\n\nLet's rewrite the given operation backwards. We are told that $$$b\\_i = a\\_i - a\\_{i-1}$$$. By rearranging this, we get $$$a\\_i = a\\_{i-1} + b\\_i$$$. If we define $$$a\\_0 = 0$$$, the array $$$a$$$ is essentially just the prefix sums of some permutation of the array $$$b$$$.\n\nWe are asked to find a permutation of $$$b$$$ that produces an array $$$a$$$ satisfying two conditions: every $$$a\\_i$$$ must be strictly positive, and $$$a$$$ must be lexicographically smallest.\n\nTo construct the lexicographically smallest array $$$a$$$, we should build it greedily from left to right. Suppose we have already successfully constructed the prefix of $$$a$$$ up to $$$a\\_{i-1}$$$. To determine $$$a\\_i$$$, we need to choose an unused element from $$$b$$$ to act as $$$b\\_i$$$. Since $$$a\\_{i-1}$$$ is already fixed, minimizing $$$a\\_i$$$ is exactly the same as picking the smallest possible $$$b\\_i$$$.\n\nHowever, we can't just pick the absolute minimum available element because we must guarantee that $$$a\\_i \\\\ge 1$$$. Substituting our earlier equation, this means $$$a\\_{i-1} + b\\_i \\\\ge 1$$$, which simplifies to $$$b\\_i \\\\ge 1 - a\\_{i-1}$$$.\n\nThis gives us a clear greedy strategy. At each step $$$i$$$, we must select the smallest available element in $$$b$$$ that is greater than or equal to $$$1 - a\\_{i-1}$$$. To implement this efficiently, we can insert all elements of the given array $$$b$$$ into a multiset. We maintain a running sum variable to represent $$$a\\_{i-1}$$$ (initially $$$0$$$). At each step, we query the multiset for the `lower_bound` of $$$1 - a\\_{i-1}$$$. If the iterator returns the end of the multiset, it means no available element is large enough to keep the current value strictly positive. In this case, a valid array cannot be formed, and we immediately output $$$-1$$$. If a valid element is found, we add it to our running sum to compute $$$a\\_i$$$, store it in our answer array, and erase that element from the multiset. We repeat this process $$$n$$$ times.\n\n**Code (C++)**\n\n```\n#include \n#define int long long\nusing namespace std;\n\nvoid solve() {\n int n, cur = 0;\n cin >> n;\n multiset b;\n for(int i = 0; i < n; i++) {\n int x;\n cin >> x;\n b.insert(x);\n }\n \n vector ans(n);\n for(int i = 0; i < n; i++){\n auto it = b.lower_bound(1 - cur);\n if(it == b.end()) return void(cout << -1 << '\\n');\n cur += *it;\n b.erase(it);\n ans[i] = cur;\n }\n \n for(int i = 0; i < n; i++) cout << ans[i] << \" \\n\"[i == n - 1];\n}\n\nsigned main() {\n ios::sync_with_stdio(false);\n cin.tie(NULL);\n int t;\n cin >> t;\n while(t--) solve();\n}\n\n```\n\n**Rate the problem!**\n\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Excellent problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Good problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Average problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Bad problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Horrible problem\n\n#### [F — Whiplash](https://codeforces.com/contest/2254/problem/F)\n\n**Hint 1**\n\nTry tracing the array elements after performing two consecutive operations, say on index $$$i$$$ and then on index $$$j$$$. If you look at the set of resulting values and ignore their order, how does it compare to the set of values you would get if you only performed a single operation on index $$$j$$$?\n\n**Hint 2**\n\nAny sequence of operations results in an array that is just a permutation of either the original array $$$a$$$, or the array $$$a$$$ after exactly one operation.\n\n**Solution**\n\nLet's analyze how a single operation affects the overall XOR sum of the array. Let $$$S\\_a$$$ be the XOR sum of all elements in $$$a$$$. When we choose an index $$$i$$$, we XOR $$$n-1$$$ elements with $$$a\\_i$$$. Since $$$n$$$ is even, $$$n-1$$$ is odd. The new XOR sum of the array will be $$$S\\_a \\\\oplus (a\\_i \\\\text{ XORed } n-1 \\\\text{ times})$$$, which simplifies perfectly to $$$S\\_a \\\\oplus a\\_i$$$.\n\nNow, let's observe what happens if we perform multiple operations. Suppose we perform an operation on index $$$i$$$, followed by an operation on index $$$j$$$. After the first operation, the element at index $$$j$$$ becomes $$$a\\_j \\\\oplus a\\_i$$$, and the element at index $$$i$$$ remains $$$a\\_i$$$. All other elements $$$a\\_k$$$ become $$$a\\_k \\\\oplus a\\_i$$$.\n\nDuring the second operation, we XOR everything (except index $$$j$$$) with the current value at index $$$j$$$, which is $$$a\\_j \\\\oplus a\\_i$$$.\n\nLet's look at the resulting values:\n\n* For any other index $$$k$$$, the value becomes $$$(a\\_k \\\\oplus a\\_i) \\\\oplus (a\\_j \\\\oplus a\\_i) = a\\_k \\\\oplus a\\_j$$$.\n* For index $$$i$$$, the value becomes $$$a\\_i \\\\oplus (a\\_j \\\\oplus a\\_i) = a\\_j$$$.\n* For index $$$j$$$, the value remains $$$a\\_j \\\\oplus a\\_i$$$.\n\nIf we look at this final set of values, it is exactly the same as if we had skipped the first operation and only performed a single operation on index $$$j$$$ from the very beginning. The order of the elements is different, but the set of values is identical.\n\nThis reveals the core property of the problem: no matter how many operations you perform, the resulting array (ignoring the order of elements) will either be identical to the original array $$$a$$$, or identical to the array $$$a$$$ after exactly one operation.\n\nThis gives us a straightforward solution. First, sort both arrays. If $$$a$$$ and $$$b$$$ are already identical, we output \"YES\" (0 operations needed). If they are not identical, we need to simulate exactly one operation. Which element $$$a\\_i$$$ should we use? We know from our first observation that the target XOR sum $$$S\\_b$$$ must equal $$$S\\_a \\\\oplus a\\_i$$$. By rearranging this, the required element must be $$$a\\_i = S\\_a \\\\oplus S\\_b$$$. We compute this target value $$$x = S\\_a \\\\oplus S\\_b$$$. If $$$x$$$ does not exist in our initial array $$$a$$$, we can immediately output \"NO\". If it does exist, we find its index, apply the operation to array $$$a$$$, sort the modified array, and check if it now matches $$$b$$$.\n\n**Code (C++)**\n\n```\n#include \nusing namespace std;\n \nsigned main() {\n int t;\n cin >> t;\n while(t--) {\n int n;\n cin >> n;\n vector a(n), b(n);\n for(auto &it : a) cin >> it;\n for(auto &it : b) cin >> it;\n \n sort(a.begin(), a.end());\n sort(b.begin(), b.end());\n\n if(a == b) {\n cout << \"YES\" << endl;\n continue;\n }\n \n int x = 0;\n for(auto it : a) x ^= it;\n for(auto it : b) x ^= it;\n\t\t\n int idx = -1;\n for(int i = 0; i < n; i++) {\n if(a[i] == x) {\n idx = i;\n break;\n }\n }\n\n if(idx == -1) {\n cout << \"NO\" << endl;\n continue;\n }\n \n for(int i = 0; i < n; i++) {\n if(i == idx) continue;\n a[i] ^= x;\n }\n \n sort(a.begin(), a.end());\n cout << (a == b ? \"YES\" : \"NO\") << endl;\n }\n}\n\n```\n\n**Rate the problem!**\n\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Excellent problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Good problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Average problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Bad problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Horrible problem\n\n#### [G — Nightcrawler](https://codeforces.com/contest/2254/problem/G)\n\n**Hint 1**\n\nBecause no subset can contain branching paths, two distinct leaves can never belong to the same subset. If the tree has $$$\\\\ell$$$ leaves, you need at least $$$\\\\ell$$$ subsets. Any $$$k \\\\lt \\\\ell$$$ is impossible. For $$$k \\\\ge \\\\ell$$$, the condition is always satisfiable since you can always break paths into smaller singletons.\n\n**Hint 2**\n\nTry to build the solution bottom-up. Each leaf starts a path. When these paths meet at a parent node, the parent can only join one of them. To maximize the total score, which path's maximum should the parent node interact with?\n\n**Hint 3**\n\nIt is always optimal for the parent to join the path that currently has the smallest maximum. If the parent's value is larger, it successfully increases that path's maximum, leaving the old maximum behind as a spare value. If the parent is smaller, the path is unaffected, and the parent itself becomes the spare value that we can use if we are allowed extra subsets.\n\n**Solution**\n\nThe core restriction is that every subset must form a straight vertical chain (an ancestor-descendant path). This immediately tells us that two distinct leaves can never be placed in the same subset. Consequently, the minimum number of subsets $$$k$$$ we can ever form is exactly the number of leaves in the tree, which we will call $$$\\\\ell$$$. For any $$$k \\\\lt \\\\ell$$$, a valid partition is impossible, and we output $$$-1$$$. For $$$k \\\\ge \\\\ell$$$, it is always possible because we can just break valid chains into individual singleton nodes.\n\nLet's build the optimal chains from the bottom up. Every leaf initially forms its own chain where the maximum value is just the leaf's value. As we move up the tree, paths merge at a parent node $$$u$$$. The node $$$u$$$ must join exactly one of the chains coming from its descendants.\n\nTo maximize the overall sum of chain maximums, $$$u$$$ should greedily join the chain with the lowest current maximum. Let's say the smallest maximum among all incoming chains is $$$x$$$. If $$$a\\_u$$$ is greater than $$$x$$$, placing $$$u$$$ in this chain upgrades its maximum from $$$x$$$ to $$$a\\_u$$$. The old maximum $$$x$$$ doesn't disappear; it gets detached and becomes an extra node that we can use to form a new chain if we are allowed to use more than $$$\\\\ell$$$ subsets. Conversely, if $$$a\\_u \\\\le x$$$, placing $$$u$$$ in the chain doesn't improve the maximum at all, so $$$a\\_u$$$ itself becomes the detached spare node.\n\nThis process translates elegantly into a data structure approach. For each subtree, we can maintain the maximums of its active chains using a min-heap (a priority queue). When we process a node $$$u$$$, we merge all the heaps of its children. To do this efficiently, we always merge smaller heaps into larger heaps, a technique known as small-to-large merging. Once the children's paths are merged, we extract the absolute minimum element from $$$u$$$'s heap, compare it with $$$a\\_u$$$, push the larger of the two back into the active heap, and append the smaller one to a global list of extra available values.\n\nBy the time we finish processing the root, the root's heap will contain exactly $$$\\\\ell$$$ elements, representing the maximums of the $$$\\\\ell$$$ essential chains. The sum of these elements gives our optimal answer for exactly $$$\\\\ell$$$ subsets. To find the answer for larger values of $$$k$$$, we simply sort our global list of extra values in descending order and greedily add the largest available extra to our running total for each additional subset allowed.\n\nBecause of the small-to-large merging, each element is moved between heaps at most $$$\\\\mathcal{O}(\\\\log n)$$$ times. Each heap operation takes logarithmic time, resulting in an overall time complexity of $$$\\\\mathcal{O}(n \\\\log^2 n)$$$, which comfortably passes the time limits.\n\n**Code (C++)**\n\n```\n#include \n \nusing namespace std;\n#define ll long long\n \nvoid solve() {\n int n;\n cin >> n;\n \n vector a(n + 1);\n for(int i = 1; i <= n; i++) cin >> a[i];\n \n vector p(n + 1);\n for(int i = 2; i <= n; i++) cin >> p[i];\n \n vector, greater>> pq(n + 1);\n vector av;\n \n for(int i = n; i >= 1; i--) {\n if(pq[i].empty()) pq[i].push(a[i]);\n else {\n ll x = pq[i].top();\n pq[i].pop();\n pq[i].push(max(x, a[i]));\n av.push_back(min(x, a[i]));\n }\n \n if(i > 1) {\n int par = p[i];\n if(pq[par].size() < pq[i].size()) swap(pq[par], pq[i]);\n \n while(!pq[i].empty()) {\n pq[par].push(pq[i].top());\n pq[i].pop();\n }\n }\n }\n \n int l = pq[1].size();\n ll sum = 0;\n \n while(!pq[1].empty()) {\n sum += pq[1].top();\n pq[1].pop();\n }\n \n vector ans(n + 1, -1);\n ans[l] = sum;\n \n sort(av.rbegin(), av.rend());\n for(int i = 0; i < av.size(); i++) {\n sum += av[i];\n ans[l + i + 1] = sum;\n }\n \n for(int k = 1; k <= n; k++) cout << ans[k] << \" \\n\"[k == n];\n}\n \nint main() {\n int t;\n cin >> t;\n while(t--) solve();\n}\n\n\n```\n\n**Rate the problem!**\n\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Excellent problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Good problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Average problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Bad problem\n* [ ](https://codeforces.com/data/like?action=like \"I like this\") Horrible problem\n\n![](//codeforces.com/codeforces.org/s/66324/images/icons/paperclip-16x16.png) Tutorial of [Codeforces Round 1114 (Div. 3)](https://codeforces.com/contest/2254) \n\n* [![Vote: I like it](//codeforces.com/codeforces.org/s/66324/images/actions/voteup-gray.png \"Vote: I like it\")](#)\n* +94\n* [![Vote: I do not like it](//codeforces.com/codeforces.org/s/66324/images/actions/votedown-gray.png \"Vote: I do not like it\")](#)\n\n* [![Author](//codeforces.com/codeforces.org/s/66324/images/blog/user_16x16.png \"Author\")](https://codeforces.com/profile/yse) [yse ](https://codeforces.com/profile/yse)\n* ![Publication date](//codeforces.com/codeforces.org/s/66324/images/blog/date_16x16.png \"Publication date\") 7 weeks ago\n* [![Comments](//codeforces.com/codeforces.org/s/66324/images/blog/comments_16x16.png \"Comments\")](https://codeforces.com/blog/entry/155666#comments) [157 ](https://codeforces.com/blog/entry/155666#comments)\n\n \n![Comments](//codeforces.com/codeforces.org/s/66324/images/icons/comments-48x48.png \"Comments\") Comments (157) \n\n[Write comment?](#) \n\n| » [bitwiseand](https://codeforces.com/profile/bitwiseand \"Newbie bitwiseand\") | 6 weeks ago, [show (+1)](#) # |\n| ----------------------------------------------------------------------------- | ----------------------------- |\n\n| » [ ![](https://codeforces.com/userpic.codeforces.org/no-avatar.jpg) ](https://codeforces.com/profile/bitwiseand) [bitwiseand](https://codeforces.com/profile/bitwiseand \"Newbie bitwiseand\") | 6 weeks ago, [hide](#) [#](https://codeforces.com/blog/entry/155666?#comment-1382713 \"Link to comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/66324/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) +3 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/66324/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) Guys is there an alternative sol for D [→](#) [Reply](#) |\n| --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ |\n\n* | » » [BenjaminNguyen](https://codeforces.com/profile/BenjaminNguyen \"Specialist BenjaminNguyen\") | 6 weeks ago, [show](#) # |\n| ----------------------------------------------------------------------------------------------- | ------------------------ |\n\n| » » [ ![](https://codeforces.com/userpic.codeforces.org/5891261/avatar/70e8b43e379920ea.jpg) ](https://codeforces.com/profile/BenjaminNguyen) [BenjaminNguyen](https://codeforces.com/profile/BenjaminNguyen \"Specialist BenjaminNguyen\") | 6 weeks ago, [hide](#) [#](https://codeforces.com/blog/entry/155666?#comment-1382884 \"Link to comment\") [^](#comment-1382713 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/66324/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/66324/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) basically what the editorial means is they take the sum of one particullar number divided by the amount of number smaller than it, if its not divisible then its impossible to create an array, if not they they need to replace the smaller number to the larger number divided by the amount of smaller one, reminder that it has to be larger than the previous replaced number [→](#) [Reply](#) |\n| ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n* | » » [adminatronik](https://codeforces.com/profile/adminatronik \"Candidate Master adminatronik\") | 5 weeks ago, [show](#) # |\n| ----------------------------------------------------------------------------------------------- | ------------------------ |\n\n| » » [ ![](https://codeforces.com/userpic.codeforces.org/1979421/avatar/8b401c75b5a8b06c.jpg) ](https://codeforces.com/profile/adminatronik) [adminatronik](https://codeforces.com/profile/adminatronik \"Candidate Master adminatronik\") | 5 weeks ago, [hide](#) [#](https://codeforces.com/blog/entry/155666?#comment-1386103 \"Link to comment\") [^](#comment-1382713 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/66324/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/66324/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) My solution: [387315277](https://codeforces.com/contest/2254/submission/387315277 \"Submission 387315277 by adminatronik\") [→](#) [Reply](#) |\n| --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n| » [Petya123](https://codeforces.com/profile/Petya123 \"Newbie Petya123\") | 6 weeks ago, [show](#) # |\n| ----------------------------------------------------------------------- | ------------------------ |\n\n| » [ ![](https://codeforces.com/userpic.codeforces.org/3566128/avatar/579a84f7ff0ef3bb.jpg) ](https://codeforces.com/profile/Petya123) [Petya123](https://codeforces.com/profile/Petya123 \"Newbie Petya123\") | 6 weeks ago, [hide](#) [#](https://codeforces.com/blog/entry/155666?#comment-1382717 \"Link to comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/66324/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) +7 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/66324/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) Nice, thank you [→](#) [Reply](#) |\n| ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n| » [RDXI](https://codeforces.com/profile/RDXI \"Newbie RDXI\") | 6 weeks ago, [show](#) # |\n| ----------------------------------------------------------- | ------------------------ |\n\n| » [ ![](https://codeforces.com/userpic.codeforces.org/5425801/avatar/831058fb3aa4f938.jpg) ](https://codeforces.com/profile/RDXI) [RDXI](https://codeforces.com/profile/RDXI \"Newbie RDXI\") | 6 weeks ago, [hide](#) [#](https://codeforces.com/blog/entry/155666?#comment-1382719 \"Link to comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/66324/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) +17 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/66324/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) Excellent contest. [→](#) [Reply](#) |\n\n...297953 bytes truncated...\n\nThe tool call succeeded but the output was truncated. Full output saved to: /Users/zayd/.local/share/opencode/tool-output/tool_0b23a686e001z7hOqcsFuczr61\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_0b23a686e001z7hOqcsFuczr61" }, "title": "https://codeforces.com/blog/entry/155666 (text/markdown; charset=utf-8)", "time": { "start": 1789696557902, "end": 1789696567408 } }, "metadata": { "openai": { "itemId": "fc_0a44c66f52a6dd5b016aac9a2d20a087d0b135a25d13434f27" } }, "id": "prt_0b23a407f001sqoabC14dggje0", "sessionID": "ses_f4dc5c871ffeijw4TBX8KF336h", "messageID": "msg_0b23a3825001URCaZVf4A1Iyie" }, { "reason": "tool-calls", "type": "step-finish", "tokens": { "total": 9275, "input": 9243, "output": 32, "reasoning": 0, "cache": { "write": 0, "read": 0 } }, "cost": 0, "id": "prt_0b23a6880001AStNDfP6q9NP8V", "sessionID": "ses_f4dc5c871ffeijw4TBX8KF336h", "messageID": "msg_0b23a3825001URCaZVf4A1Iyie" } ] }, { "info": { "parentID": "msg_0b23a37a1001LUqDes84ANGaSC", "role": "assistant", "mode": "nudge-generation", "agent": "nudge-generation", "variant": "low", "path": { "cwd": "/private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-XtZbAE", "root": "/" }, "cost": 0, "tokens": { "total": 23072, "input": 13828, "output": 242, "reasoning": 42, "cache": { "write": 0, "read": 8960 } }, "modelID": "gpt-6-astra", "providerID": "openai", "time": { "created": 1789696567430, "completed": 1789696578582 }, "finish": "tool-calls", "id": "msg_0b23a6886001fWDS6C6yhvcOJR", "sessionID": "ses_f4dc5c871ffeijw4TBX8KF336h" }, "parts": [ { "type": "step-start", "id": "prt_0b23a71f000101XXumqw9EPtyd", "sessionID": "ses_f4dc5c871ffeijw4TBX8KF336h", "messageID": "msg_0b23a6886001fWDS6C6yhvcOJR" }, { "type": "reasoning", "text": "**Verifying sample length**", "time": { "start": 1789696569842, "end": 1789696573112 }, "metadata": { "openai": { "itemId": "rs_04e2ad531fe81f64016aac9a39c85887d0b21e8b5b5a941522", "reasoningEncryptedContent": "gAAAAABqrJo8ghiut5H-i03u9NMmvdBkky7u1NLp6jHM_cGSzagY04AiDL-aqZGUkyTF6vVmxSXLFJ5AobH7Ict868ZFqPLsk7oQPjSKIwiVZYKOaViYjp1SIM_d-mvemqvCFV1XlJ2JFQ57G2gVK7yLKyNbRZDdy0Ceyq3-fFautqONOe_vHCEoMo5u0bh8CoN_kEPiNZJczMi_7GSKH_ahDabHoWf6Sad3E4vSZhUgMqNsD22NXtwc7IcUCKpIQq1WjLKpTg9o6b4dsdFHbxb1PDrLY-BHKPyIMJYwAQQtqOFmZ12-k1tS5x37pnlTVcP4G6zvvFvoQTpOxZgxYtyCSKtxrXg01JsEkNCkUUSIKNnjINKjrUVM943-pI-KS29AephikYumscZZiXp2AH__JUgASUFxujCpTMnZCIxdyVfAQhS8ggYr8HU1x9VQpX_NdlXE0CdIC5KanpdabnewuqHrvU9mvhnOVmQw2Gx4USPhagkMlbq9RWsOGbvgGlPj7Q72tQFNxtd_p4DvUz93z6oRzK4sYpHUIAj6Vcvsl90fNFZQ6eQ49naP_LOC4uORRqC2qkdKZirEal0he1PPbi_8za4kA2Dv1XfHn1jrA2KShVveWqybJotuqKMH11NDUuJEIPehMpKm8aY_ItRTrW4W2xZwqxHTg1eyKsw8M2U4VxU_yXueRpcTp92aq5ibMooYhxFXzoz3m6kLFYLQHEXYx7gdCRe0gRDMhuIknK20_-t_K9kPhas0L_7fAvGMzCZAW3AMPj2d1aYN-DSnvccFNETfUPaMZggGkHrty9lNdEAE83uU7xBfEvkDpy_aESXCiHf1jNehVwPsFdsNkZmGUlX_n5HCVQM4ZwKpXUrd_SsJEOPghjucWvBUYAch6JuimBjkQ5ox6dE9Nhl5lXLwLOSFFzwDKsUcd32TXyM9oZ_Q-WR313L5NdYtp6k7LikFjC6luJ9UpGiBR_71KdLQKotJ2SSCekcK-KD7FvUYV-CMFtxlwQ-8_BOsc5E4pLBEyiuWtCBmF7UgDEXQDyXGAxfo6ol4bIsrf_iDdATryGw2Fe5AHi5nQJnUFJsfvvXbh50kpgPtvz2swSILOdbc4A_wNfNyGc1Ah1zHapVrO7Ce3N7CSaz_Nf3zzypxuqK5zHMXqrUbeB-MSkk41JCKvIupjmGQRldgmVpZNqmeggtUQg49UiHE2w9iiI0bQj-TklkXNCAiQjNUNIk6cYlrSZsdX20BpBwb_goj0S36k0AXisgmgkKUUpFTIdG2YHMJnUHCxQSdbQZZ1Slpzti4QmUa2exaFG0fP2378stbXdzvDj0vsFCOlBiKQ0idRd1pfgkd5UOWO71G-ZASIwBp4ap6Dy8kYK_trmumzShyUTe31KoDcttqZvSWm4nTvU6laQVJsg-h4aBTRinEtNZutDWDmnhPvfTxnPrblPPSu-vP1bWOY19Xr9MMg9onOT7x4ydX" } }, "id": "prt_0b23a71f2001VJ93TlpBu7SQrJ", "sessionID": "ses_f4dc5c871ffeijw4TBX8KF336h", "messageID": "msg_0b23a6886001fWDS6C6yhvcOJR" }, { "type": "tool", "tool": "bash", "callID": "call_tIAj1ym3hVmOeO2HZuzH1iSU", "state": { "status": "completed", "input": { "command": "python3 -c 'from itertools import product\ncount = 0\nfor n in range(3, 9):\n for chars in product(\"abc\", repeat=n):\n s = \"\".join(chars)\n runs = 1 + sum(s[i] != s[i-1] for i in range(1,n))\n gain = 0\n for i in range(1,n-1):\n if s[i] != s[i-1] and s[i] != s[i+1]:\n gain = max(gain, 2 if s[i-1] == s[i+1] else 1)\n actual = min(1 + sum(a != b for a,b in zip(u,u[1:])) for i in range(1,n-1) for u in [s[:i]+s[i+1:]])\n assert runs-gain == actual, s\n count += 1\nprint(\"Verified\", count, \"strings against exhaustive legal deletions.\")'" }, "output": "Verified 9828 strings against exhaustive legal deletions.\n", "metadata": { "output": "Verified 9828 strings against exhaustive legal deletions.\n", "exit": 0, "truncated": false }, "title": "python3 -c 'from itertools import product\ncount = 0\nfor n in range(3, 9):\n for chars in product(\"abc\", repeat=n):\n s = \"\".join(chars)\n runs = 1 + sum(s[i] != s[i-1] for i in range(1,n))\n gain = 0\n for i in range(1,n-1):\n if s[i] != s[i-1] and s[i] != s[i+1]:\n gain = max(gain, 2 if s[i-1] == s[i+1] else 1)\n actual = min(1 + sum(a != b for a,b in zip(u,u[1:])) for i in range(1,n-1) for u in [s[:i]+s[i+1:]])\n assert runs-gain == actual, s\n count += 1\nprint(\"Verified\", count, \"strings against exhaustive legal deletions.\")'", "time": { "start": 1789696578501, "end": 1789696578580 } }, "metadata": { "openai": { "itemId": "fc_04e2ad531fe81f64016aac9a3cf9b487d08033e9498c830664" } }, "id": "prt_0b23a7eba0010CGq6m6kbrO87O", "sessionID": "ses_f4dc5c871ffeijw4TBX8KF336h", "messageID": "msg_0b23a6886001fWDS6C6yhvcOJR" }, { "reason": "tool-calls", "type": "step-finish", "tokens": { "total": 23072, "input": 13828, "output": 242, "reasoning": 42, "cache": { "write": 0, "read": 8960 } }, "cost": 0, "id": "prt_0b23a9415001sAVp3wLUoV7c7q", "sessionID": "ses_f4dc5c871ffeijw4TBX8KF336h", "messageID": "msg_0b23a6886001fWDS6C6yhvcOJR" } ] }, { "info": { "parentID": "msg_0b23a37a1001LUqDes84ANGaSC", "role": "assistant", "mode": "nudge-generation", "agent": "nudge-generation", "variant": "low", "path": { "cwd": "/private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-XtZbAE", "root": "/" }, "cost": 0, "tokens": { "total": 24481, "input": 566, "output": 1387, "reasoning": 0, "cache": { "write": 0, "read": 22528 } }, "modelID": "gpt-6-astra", "providerID": "openai", "time": { "created": 1789696578583, "completed": 1789696625067 }, "finish": "tool-calls", "structured": { "status": "success", "reason": null, "hints": [ { "order": 1, "content": "You only need the **length** of the compressed string. What does that length count in the original string?" }, { "order": 2, "content": "It counts maximal blocks of equal characters. If you delete a character from a block of length at least $2$, that block still exists, so the number of blocks stays the same." }, { "order": 3, "content": "The only way to improve the answer is to remove an entire block of length $1$. For an internal position $i$, this happens exactly when $s_i \\ne s_{i-1}$ and $s_i \\ne s_{i+1}$." }, { "order": 4, "content": "After removing a singleton block, its neighboring blocks touch. If their characters differ, you lose one block. If their characters match, they merge too, so you lose two blocks." }, { "order": 5, "content": "Count the original blocks as $R=1+\\sum_{i=2}^{n}[s_i\\ne s_{i-1}]$. Scan only $2\\le i\\le n-1$ and find the largest possible reduction: $0$ for a non-singleton, $1$ for a singleton with different neighbors, or $2$ for a singleton with equal neighbors. Output $R$ minus that maximum. A reduction of $0$ still means you perform the required deletion—it just doesn't change the compressed length." } ], "editorial": "The compressed length is simply the **number of maximal blocks of equal characters**. For example, `aabbbaa` has three blocks: `aa | bbb | aa`.\n\nSo the question is: how many blocks can one legal deletion remove?\n\n### What a deletion changes\n\nConsider deleting an internal character $s_i$.\n\n**1. Its block has length at least $2$.**\n\nThe block gets shorter, but it doesn't disappear. Its neighboring blocks remain separated, so the compressed length stays unchanged.\n\nFor example, deleting either `b` from `abba` produces `aba`. Both strings have three blocks. Fewer characters does **not** automatically mean fewer blocks.\n\n**2. Its block has length $1$.**\n\nThe block disappears, and the blocks on either side become adjacent:\n\n- If their characters differ, they stay separate. The number of blocks decreases by **$1$**. Example: `abc` becomes `ac`.\n- If their characters match, they merge into one block. The number of blocks decreases by **$2$**: one for the deleted block and one for the merge. Example: `abaa` becomes `aaa`, going from three blocks to one.\n\nNo other blocks change. That gives us all possible outcomes: a reduction of $0$, $1$, or $2$.\n\n### Recognizing singleton blocks\n\nThere is no need to build the compressed string or store block lengths. An internal character is a singleton block exactly when it differs from both immediate neighbors:\n\n$$s_i\\ne s_{i-1}\\quad\\text{and}\\quad s_i\\ne s_{i+1}.$$\n\nIf this holds, compare $s_{i-1}$ and $s_{i+1}$ to decide whether the reduction is $1$ or $2$.\n\n### Algorithm\n\nFor each test case:\n\n1. Count the original number of blocks $R$. Start at $1$, then add $1$ whenever two consecutive characters differ.\n2. Initialize `bestReduction = 0`.\n3. Check every internal position. If it is a singleton block, update the best reduction with $2$ when its neighbors match, or $1$ otherwise.\n4. Output $R-\\text{bestReduction}$.\n\nIn zero-based C++ indexing, legal deletion positions are `1` through `n - 2`. Never consider the endpoints, even if they form singleton blocks.\n\n**Exactly one deletion still matters.** If the best reduction is $0$, we aren't skipping the operation. Since $n\\ge3$, a legal position exists, and in this case deleting any internal character leaves the number of blocks unchanged. An all-equal string, for example, still has answer $1$.\n\n### Correctness proof\n\nLet $R$ be the original number of blocks.\n\nFor any legal deletion, either the deleted character belongs to a block of length at least $2$, or it is a singleton block.\n\n- In the first case, its block survives and the reduction is $0$.\n- In the second case, its block disappears. The neighboring blocks merge exactly when their characters match, giving a reduction of $2$ if they match and $1$ otherwise.\n\nThe algorithm checks every legal position and computes its reduction according to these exhaustive cases. Therefore, `bestReduction` is the largest reduction achievable by a legal deletion. Subtracting it from $R$ gives the minimum possible compressed length.\n\n### Complexity\n\nBoth scans take $O(n)$ time per test case, or $O(\\sum n)$ overall. The algorithm uses $O(1)$ auxiliary space beyond the $O(n)$ input string.", "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 string s;\n cin >> n >> s;\n\n int runs = 1;\n for (int i = 1; i < n; ++i) {\n runs += (s[i] != s[i - 1]);\n }\n\n int bestReduction = 0;\n for (int i = 1; i + 1 < n; ++i) {\n if (s[i] != s[i - 1] && s[i] != s[i + 1]) {\n int reduction = (s[i - 1] == s[i + 1] ? 2 : 1);\n bestReduction = max(bestReduction, reduction);\n }\n }\n\n cout << runs - bestReduction << '\\n';\n }\n}\n" }, "id": "msg_0b23a9417001w1ETyVF0w7zqQs", "sessionID": "ses_f4dc5c871ffeijw4TBX8KF336h" }, "parts": [ { "type": "step-start", "id": "prt_0b23aa7fb001DI84DeMyXjkAD8", "sessionID": "ses_f4dc5c871ffeijw4TBX8KF336h", "messageID": "msg_0b23a9417001w1ETyVF0w7zqQs" }, { "type": "tool", "tool": "StructuredOutput", "callID": "call_5VZSpRgt8EqExhXIIlU460EU", "state": { "status": "completed", "input": { "status": "success", "reason": null, "hints": [ { "order": 1, "content": "You only need the **length** of the compressed string. What does that length count in the original string?" }, { "order": 2, "content": "It counts maximal blocks of equal characters. If you delete a character from a block of length at least $2$, that block still exists, so the number of blocks stays the same." }, { "order": 3, "content": "The only way to improve the answer is to remove an entire block of length $1$. For an internal position $i$, this happens exactly when $s_i \\ne s_{i-1}$ and $s_i \\ne s_{i+1}$." }, { "order": 4, "content": "After removing a singleton block, its neighboring blocks touch. If their characters differ, you lose one block. If their characters match, they merge too, so you lose two blocks." }, { "order": 5, "content": "Count the original blocks as $R=1+\\sum_{i=2}^{n}[s_i\\ne s_{i-1}]$. Scan only $2\\le i\\le n-1$ and find the largest possible reduction: $0$ for a non-singleton, $1$ for a singleton with different neighbors, or $2$ for a singleton with equal neighbors. Output $R$ minus that maximum. A reduction of $0$ still means you perform the required deletion—it just doesn't change the compressed length." } ], "editorial": "The compressed length is simply the **number of maximal blocks of equal characters**. For example, `aabbbaa` has three blocks: `aa | bbb | aa`.\n\nSo the question is: how many blocks can one legal deletion remove?\n\n### What a deletion changes\n\nConsider deleting an internal character $s_i$.\n\n**1. Its block has length at least $2$.**\n\nThe block gets shorter, but it doesn't disappear. Its neighboring blocks remain separated, so the compressed length stays unchanged.\n\nFor example, deleting either `b` from `abba` produces `aba`. Both strings have three blocks. Fewer characters does **not** automatically mean fewer blocks.\n\n**2. Its block has length $1$.**\n\nThe block disappears, and the blocks on either side become adjacent:\n\n- If their characters differ, they stay separate. The number of blocks decreases by **$1$**. Example: `abc` becomes `ac`.\n- If their characters match, they merge into one block. The number of blocks decreases by **$2$**: one for the deleted block and one for the merge. Example: `abaa` becomes `aaa`, going from three blocks to one.\n\nNo other blocks change. That gives us all possible outcomes: a reduction of $0$, $1$, or $2$.\n\n### Recognizing singleton blocks\n\nThere is no need to build the compressed string or store block lengths. An internal character is a singleton block exactly when it differs from both immediate neighbors:\n\n$$s_i\\ne s_{i-1}\\quad\\text{and}\\quad s_i\\ne s_{i+1}.$$\n\nIf this holds, compare $s_{i-1}$ and $s_{i+1}$ to decide whether the reduction is $1$ or $2$.\n\n### Algorithm\n\nFor each test case:\n\n1. Count the original number of blocks $R$. Start at $1$, then add $1$ whenever two consecutive characters differ.\n2. Initialize `bestReduction = 0`.\n3. Check every internal position. If it is a singleton block, update the best reduction with $2$ when its neighbors match, or $1$ otherwise.\n4. Output $R-\\text{bestReduction}$.\n\nIn zero-based C++ indexing, legal deletion positions are `1` through `n - 2`. Never consider the endpoints, even if they form singleton blocks.\n\n**Exactly one deletion still matters.** If the best reduction is $0$, we aren't skipping the operation. Since $n\\ge3$, a legal position exists, and in this case deleting any internal character leaves the number of blocks unchanged. An all-equal string, for example, still has answer $1$.\n\n### Correctness proof\n\nLet $R$ be the original number of blocks.\n\nFor any legal deletion, either the deleted character belongs to a block of length at least $2$, or it is a singleton block.\n\n- In the first case, its block survives and the reduction is $0$.\n- In the second case, its block disappears. The neighboring blocks merge exactly when their characters match, giving a reduction of $2$ if they match and $1$ otherwise.\n\nThe algorithm checks every legal position and computes its reduction according to these exhaustive cases. Therefore, `bestReduction` is the largest reduction achievable by a legal deletion. Subtracting it from $R$ gives the minimum possible compressed length.\n\n### Complexity\n\nBoth scans take $O(n)$ time per test case, or $O(\\sum n)$ overall. The algorithm uses $O(1)$ auxiliary space beyond the $O(n)$ input string.", "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 string s;\n cin >> n >> s;\n\n int runs = 1;\n for (int i = 1; i < n; ++i) {\n runs += (s[i] != s[i - 1]);\n }\n\n int bestReduction = 0;\n for (int i = 1; i + 1 < n; ++i) {\n if (s[i] != s[i - 1] && s[i] != s[i + 1]) {\n int reduction = (s[i - 1] == s[i + 1] ? 2 : 1);\n bestReduction = max(bestReduction, reduction);\n }\n }\n\n cout << runs - bestReduction << '\\n';\n }\n}\n" }, "output": "Structured output captured successfully.", "metadata": { "valid": true }, "title": "Structured Output", "time": { "start": 1789696625012, "end": 1789696625015 } }, "metadata": { "openai": { "itemId": "fc_07c536d84a268e51016aac9a479b0887d0bf7ac3d71cd00dca" } }, "id": "prt_0b23aa7fd001C56N7mM15hNsff", "sessionID": "ses_f4dc5c871ffeijw4TBX8KF336h", "messageID": "msg_0b23a9417001w1ETyVF0w7zqQs" }, { "reason": "tool-calls", "type": "step-finish", "tokens": { "total": 24481, "input": 566, "output": 1387, "reasoning": 0, "cache": { "write": 0, "read": 22528 } }, "cost": 0, "id": "prt_0b23b49a9001EUG7Bp545cs7iR", "sessionID": "ses_f4dc5c871ffeijw4TBX8KF336h", "messageID": "msg_0b23a9417001w1ETyVF0w7zqQs" } ] } ] }