{
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    "slug": "gentle-island",
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    "path": "private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-DiYqcB",
    "title": "Nudge content generation",
    "agent": "nudge-generation",
    "model": {
      "id": "gpt-5.6-sol",
      "providerID": "openai",
      "variant": "max"
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      "reasoning": 3488,
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  "messages": [
    {
      "info": {
        "role": "user",
        "time": {
          "created": 1784319825628
        },
        "agent": "nudge-generation",
        "model": {
          "providerID": "openai",
          "modelID": "gpt-5.6-sol",
          "variant": "max"
        },
        "system": "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"
                  },
                  {
                    "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",
                    "description": "Exactly 5 progressive hints, each building on the last.",
                    "items": {
                      "type": "object",
                      "properties": {
                        "order": {
                          "type": "number",
                          "description": "Hint number, 1 through 5."
                        },
                        "content": {
                          "type": "string",
                          "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"
                  },
                  {
                    "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"
                  },
                  {
                    "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_f71bfd6dc001exk58JZEOmZu8u",
        "sessionID": "ses_08e402936ffeAiEC3Va73qZZG4"
      },
      "parts": [
        {
          "type": "text",
          "text": "Generate Codeforces learning content: five progressive hints, an original editorial, and AC-quality C++.\n\nSolve from the supplied statement first. For very hard problems, do a quick source check for the official Codeforces tutorial/editorial and accepted submissions when they would help you derive or verify the solution. If a source is blocked, missing, Cloudflare-challenged, 404, or otherwise unavailable, keep solving from the supplied statement and remember that source status. If you still can't solve after that, it's ok.\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 2244E: \"Masha and the Garland\".\n\nKnown problem metadata:\n- Contest ID: 2244\n- Index: E\n- Name: Masha and the Garland\n- Rating: unknown / unrated\n- Tags: data structures, dp, strings\n- Problem URL: https://codeforces.com/contest/2244/problem/E\n\nUse the rating and tags as weak signals only. The supplied statement is the source of truth.\n\nProblem Statement:\n<problem-statement>\n<div class=\"header\"><div class=\"title\">E. Masha and the Garland</div><div class=\"time-limit\"><div class=\"property-title\">time limit per test</div>2 seconds</div><div class=\"memory-limit\"><div class=\"property-title\">memory limit per test</div>256 megabytes</div><div class=\"input-file input-standard\"><div class=\"property-title\">input</div>standard input</div><div class=\"output-file output-standard\"><div class=\"property-title\">output</div>standard output</div></div><div><p>Masha has a New Year garland consisting of $$$n$$$ bulbs. Each bulb can be either off or on. The state of the garland is given by a binary string $$$s$$$ of length $$$n$$$, where '<span class=\"tex-font-style-tt\">0</span>' denotes an off bulb and '<span class=\"tex-font-style-tt\">1</span>' denotes an on bulb.</p><p>Masha considers the garland <span class=\"tex-font-style-it\">beautiful</span> if the states of adjacent bulbs strictly alternate. That is, there are no two adjacent bulbs that are both on or both off. For example, the garlands '<span class=\"tex-font-style-tt\">01010</span>' and '<span class=\"tex-font-style-tt\">1010</span>' are beautiful, while '<span class=\"tex-font-style-tt\">0110</span>' and '<span class=\"tex-font-style-tt\">000</span>' are not.</p><p>Yura can apply the following operation to the garland: choose a subsegment and flip the state of all bulbs in it, that is, turn all on bulbs off and all off bulbs on.</p><p>Masha invites him to play the following game $$$q$$$ times: she chooses a segment from the $$$l$$$-th to the $$$r$$$-th bulb inclusive, and Yura must make this segment <span class=\"tex-font-style-it\">beautiful</span> using at most $$$k$$$ operations.</p><p>However, Yura is not sure whether this is always possible, so he asks you to determine for each game whether he can make the chosen segment <span class=\"tex-font-style-it\">beautiful</span> using no more than $$$k$$$ operations. Note that the games are independent, and the garland itself is not actually modified.</p></div><div class=\"input-specification\"><div class=\"section-title\">Input</div><p>The first line contains a single integer $$$t$$$ ($$$1 \\le t \\le 10^4$$$) — the number of test cases.</p><p>The first line of each test case contains two integers $$$n$$$ and $$$q$$$ ($$$1 \\le n, q \\le 2 \\cdot 10^5$$$) — the length of the garland and the number of games.</p><p>The second line contains a binary string $$$s$$$ of length $$$n$$$, consisting only of characters '<span class=\"tex-font-style-tt\">0</span>' and '<span class=\"tex-font-style-tt\">1</span>'.</p><p>The next $$$q$$$ lines each contain three integers $$$l$$$, $$$r$$$, and $$$k$$$ ($$$1 \\le l \\le r \\le n$$$, $$$0 \\le k \\le n$$$) — the boundaries of the segment and the maximum allowed number of operations.</p><p>It is guaranteed that the sum of $$$n$$$ and the sum of $$$q$$$ over all test cases do not exceed $$$2 \\cdot 10^5$$$.</p></div><div class=\"output-specification\"><div class=\"section-title\">Output</div><p>For each game, output \"<span class=\"tex-font-style-tt\">YES</span>\" if the segment can be made beautiful using at most $$$k$$$ operations, and \"<span class=\"tex-font-style-tt\">NO</span>\" otherwise.</p></div><div class=\"sample-tests\"><div class=\"section-title\">Example</div><div class=\"sample-test\"><div class=\"input\"><div class=\"title\">Input</div><pre><div class=\"test-example-line test-example-line-even test-example-line-0\">2</div><div class=\"test-example-line test-example-line-odd test-example-line-1\">5 5</div><div class=\"test-example-line test-example-line-odd test-example-line-1\">00110</div><div class=\"test-example-line test-example-line-odd test-example-line-1\">1 5 1</div><div class=\"test-example-line test-example-line-odd test-example-line-1\">1 5 2</div><div class=\"test-example-line test-example-line-odd test-example-line-1\">2 4 1</div><div class=\"test-example-line test-example-line-odd test-example-line-1\">1 2 0</div><div class=\"test-example-line test-example-line-odd test-example-line-1\">3 4 0</div><div class=\"test-example-line test-example-line-even test-example-line-2\">4 2</div><div class=\"test-example-line test-example-line-even test-example-line-2\">1010</div><div class=\"test-example-line test-example-line-even test-example-line-2\">1 4 0</div><div class=\"test-example-line test-example-line-even test-example-line-2\">2 3 1</div></pre></div><div class=\"output\"><div class=\"title\">Output</div><pre><div class=\"test-example-line test-example-line-odd test-example-line-1\">YES</div><div class=\"test-example-line test-example-line-odd test-example-line-1\">YES</div><div class=\"test-example-line test-example-line-odd test-example-line-1\">YES</div><div class=\"test-example-line test-example-line-odd test-example-line-1\">NO</div><div class=\"test-example-line test-example-line-odd test-example-line-1\">NO</div><div class=\"test-example-line test-example-line-even test-example-line-2\">YES</div><div class=\"test-example-line test-example-line-even test-example-line-2\">YES</div></pre></div></div></div>\n</problem-statement>\n\n\nSource lookup status:\n<source-lookup-status>\n- Tutorial: https://codeforces.com/blog/entry/155197 loaded (200 OK; title: \"Codeforces Round 1109 (Div. 3) Editorial - Codeforces\"; mentions 2244E)\n</source-lookup-status>\n\nGenerate:\n1. Five progressive hints, from a gentle nudge to the key insight.\n2. A deep editorial explaining like literally everything.\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- 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 JUST the hint text. No \"Hint 1:\" or subtitles. The UI adds those automatically.\n- Do not start the editorial with an \"# Editorial\" heading. The UI already adds that section. Feel free to add other headers as needed though.\n- Use styling in the editorial if needed! Just that first header is no good but any subheaders or bolding or whatever is good!\n\nFor the C++ solution, you MUST use this template and work around it:\n\n```cpp\n#include <bits/stdc++.h>\nusing namespace std;\n\nusing ll = long long;\n\nvoid setIO() {\n    ios::sync_with_stdio(false);\n    cin.tie(nullptr);\n}\n\nint main() {\n    setIO();\n}\n```",
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        "mode": "nudge-generation",
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        "variant": "max",
        "path": {
          "cwd": "/private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-DiYqcB",
          "root": "/"
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        "providerID": "openai",
        "time": {
          "created": 1784319825924,
          "completed": 1784319834992
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        {
          "type": "reasoning",
          "text": "**Fetching official editorial**",
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            "start": 1784319827394,
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          "callID": "call_itTRcaeSuyynPG4MGqBhsaJv",
          "state": {
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            "input": {
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              "format": "markdown",
              "timeout": 30
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            "output": "---\nimage: //codeforces.com/codeforces.org/s/90960/images/codeforces-sponsored-by-ton.png\ntitle: Codeforces Round 1109 (Div. 3) Editorial - Codeforces\ndescription: Codeforces. Programming competitions and contests, programming community\n---\n\n<!DOCTYPE HTML PUBLIC \"-//W3C//DTD HTML 4.01//EN\"> \n\n  \n[![Codeforces](//codeforces.com/codeforces.org/s/90960/images/codeforces-sponsored-by-ton.png \"Codeforces\")](https://codeforces.com/) \n\n[![In English](//codeforces.com/codeforces.org/s/90960/images/flags/24/gb.png \"In English\")](https://codeforces.com/blog/entry/155197?locale=en) [![По-русски](//codeforces.com/codeforces.org/s/90960/images/flags/24/ru.png \"По-русски\")](https://codeforces.com/blog/entry/155197?locale=ru) \n\n[Enter](https://codeforces.com/enter?back=%2Fblog%2Fentry%2F155197)| [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* [Edu](https://codeforces.com/edu/courses)\n* [API](https://codeforces.com/apiHelp)\n* [Calendar](https://codeforces.com/calendar)\n* [Help](https://codeforces.com/help)\n  \n  \n→ Pay attention \n\nBefore contest  \n[Codeforces Round 1111 (Div. 2)](https://codeforces.com/contests/2247)  \n18:11:12  \n[Register now »](https://codeforces.com/contestRegistration/2247)\n\n\\*has extra registration\n\n→ Top rated \n\n| # 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English](//codeforces.com/codeforces.org/s/90960/images/flags/24/gb.png \"In English\") \n\n[2244A - Iskander and Drawings](https://codeforces.com/contest/2244/problem/A \"Codeforces Round 1109 (Div. 3)\")\n\n**Tutorial**\n\nTutorial is loading...\n\n**Solution**\n\n```\n#include <bits/stdc++.h>\n\nusing ll = long long;\nusing namespace std;\n\nvoid I_love_feblokas();\n\nint32_t main() {\n    int32_t tc = 1;\n    cin >> tc;\n    while (tc-->0) {\n        I_love_feblokas();\n    }\n    return 0;\n}\n\nvoid I_love_feblokas() {\n    int32_t ans = 0, cnt = 0;\n    int n;\n    cin >> n;\n    string s;\n    cin >> s;\n    for (auto &c : s) {\n        if (c == '*') {\n            cnt = 0;\n        } else {\n            cnt++;\n        }\n        ans = max(ans, (cnt + 1) / 2);\n    }\n    cout << ans << '\\n';\n    return;\n}\n\n```\n\n[2244B - Nikita and Books](https://codeforces.com/contest/2244/problem/B \"Codeforces Round 1109 (Div. 3)\")\n\n**Tutorial**\n\nTutorial is loading...\n\n**Solution**\n\n```\n#include <bits/stdc++.h>\n\nusing ll = long long;\nusing namespace std;\n\nvoid I_love_feblokas();\n\nint32_t main() {\n    int32_t tc = 1;\n    cin >> tc;\n    while (tc-- > 0) {\n        I_love_feblokas();\n    }\n    return 0;\n}\n\nvoid I_love_feblokas() {\n    int n;\n    cin >> n;\n    vector<ll> a(n);\n    for (auto &x : a) cin >> x;\n    ll cur = 0;\n    bool ok = true;\n    for (ll i = 0; i < n; ++i) {\n        cur += a[i];\n        ll need = (i + 1) * (i + 2) / 2;\n        if (cur < need) {\n            ok = false;\n        }\n    }\n    if (ok) {\n        cout << \"YES\\n\";\n    } else {\n        cout << \"NO\\n\";\n    }\n}\n\n```\n\n[2244C - Stepan and Permutation](https://codeforces.com/contest/2244/problem/C \"Codeforces Round 1109 (Div. 3)\")\n\n**Tutorial**\n\nTutorial is loading...\n\n**Solution**\n\n```\n#include <bits/stdc++.h>\n\nusing ll = long long;\nusing namespace std;\n\nvoid I_love_feblokas();\n\nint32_t main() {\n    int32_t tc = 1;\n    cin >> tc;\n    while (tc-- > 0) {\n        I_love_feblokas();\n    }\n    return 0;\n}\n\nvoid I_love_feblokas() {\n    int n, x, y;\n    cin >> n >> x >> y;\n    vector<int> p(n);\n    for (auto &val : p) cin >> val;\n    \n    int g = gcd(x, y);\n    bool ok = true;\n    \n    for (int i = 0; i < n; ++i) {\n        if ((p[i] % g) != ((i + 1) % g)) {\n            ok = false;\n            break;\n        }\n    }\n    \n    if (ok) {\n        cout << \"YES\\n\";\n    } else {\n        cout << \"NO\\n\";\n    }\n}\n\n```\n\n[2244D - Yaroslav and Productivity](https://codeforces.com/contest/2244/problem/D \"Codeforces Round 1109 (Div. 3)\")\n\n**Tutorial**\n\nTutorial is loading...\n\n**Solution**\n\n```\n#include <bits/stdc++.h>\n\nusing ll = long long;\nusing namespace std;\n\nvoid I_love_feblokas();\n\nint32_t main() {\n    int32_t tc = 1;\n    cin >> tc;\n    while (tc-->0) {\n        I_love_feblokas();\n    }\n    return 0;\n}\n\nvoid I_love_feblokas() {\n    int n, m;\n    ll ans = 0;\n    cin >> n >> m;\n    vector<ll> a(n), b(m);\n    for (auto &x : a) cin >> x;\n    for (auto &x : b) cin >> x;\n    b.push_back(0);\n    sort(b.begin(), b.end());\n    vector<ll> pref(n + 1);\n    for (ll i = 0; i < n; ++i) {\n        pref[i + 1] = pref[i] + a[i];\n    }\n    for (ll i = 1; i < b.size(); ++i) {\n        ans += abs(pref[b[i]] - pref[b[i - 1]]);\n    }\n    ans += pref[n] - pref[b.back()];\n    cout << ans << '\\n';\n    return;\n}\n\n\n```\n\n[2244E - Masha and the Garland](https://codeforces.com/contest/2244/problem/E \"Codeforces Round 1109 (Div. 3)\")\n\n**Tutorial**\n\nTutorial is loading...\n\n**Solution**\n\n```\n#include <bits/stdc++.h>\n\nusing namespace std;\n\nvoid solve() {\n    int n, q;\n    cin >> n >> q;\n    string s;\n    cin >> s;\n    vector<int> pref(n, 0);\n    for (int i = 0; i < n &mdash; 1; ++i) {\n        pref[i + 1] = pref[i] + (s[i] == s[i + 1] ? 1 : 0);\n    }\n    for (int i = 0; i < q; ++i) {\n        int l, r, k;\n        cin >> l >> r >> k;\n        if (l == r) {\n            cout << \"YES\\n\";\n            continue;\n        }\n        int c = pref[r &mdash; 1] &mdash; pref[l &mdash; 1];\n        int needed = (c + 1) / 2;\n        if (needed <= k) cout << \"YES\\n\";\n        else cout << \"NO\\n\";\n    }\n}\n\nint main() {\n    int t;\n    cin >> t;\n    while (t--) solve();\n    return 0;\n}\n\n```\n\n[2244F - Anya Loves Trees!](https://codeforces.com/contest/2244/problem/F \"Codeforces Round 1109 (Div. 3)\")\n\n**Tutorial**\n\nTutorial is loading...\n\n**Solution**\n\n```\n#include <bits/stdc++.h>\n\nusing ll = long long;\nusing namespace std;\n\nvector<vector<int>> g;\nvector<int> leaf;\nbool ok;\n\nvoid I_love_feblokas();\npair<int, int> dfs(int u, int p);\n\nint32_t main() {\n    int32_t tc = 1;\n    cin >> tc;\n    while (tc-- > 0) {\n        I_love_feblokas();\n    }\n    return 0;\n}\n\npair<int, int> dfs(int u, int p) {\n    if (!ok) return {0, 0};\n    if (leaf[u] != 0) {\n        for (int v : g[u]) {\n            if (v != p) {\n                ok = false;\n                return {0, 0};\n            }\n        }\n        return {leaf[u], leaf[u]};\n    }\n    vector<pair<int, int>> segs;\n    for (int v : g[u]) {\n        if (v != p) {\n            segs.push_back(dfs(v, u));\n            if (!ok) return {0, 0};\n        }\n    }\n    if (segs.empty()) {\n        ok = false;\n        return {0, 0};\n    }\n    vector<pair<int, int>> sorted_segs = segs;\n    sort(sorted_segs.begin(), sorted_segs.end());\n    for (size_t i = 0; i < sorted_segs.size() &mdash; 1; ++i) {\n        if (sorted_segs[i].second + 1 != sorted_segs[i + 1].first) {\n            ok = false;\n            return {0, 0};\n        }\n    }\n    int start_pos = -1;\n    for (size_t i = 0; i < segs.size(); ++i) {\n        if (segs[i] == sorted_segs[0]) {\n            start_pos = i;\n            break;\n        }\n    }\n    if (start_pos == -1) {\n        ok = false;\n        return {0, 0};\n    }\n    for (size_t i = 0; i < segs.size(); ++i) {\n        if (segs[(start_pos + i) % segs.size()] != sorted_segs[i]) {\n            ok = false;\n            return {0, 0};\n        }\n    }\n    return {sorted_segs[0].first, sorted_segs.back().second};\n}\n\nvoid I_love_feblokas() {\n    int n;\n    cin >> n;\n    g.assign(n + 1, vector<int>());\n    leaf.assign(n + 1, 0);\n    ok = true;\n    for (int i = 2; i <= n; ++i) {\n        int p;\n        cin >> p;\n        g[p].push_back(i);\n        g[i].push_back(p);\n    }\n    int k = 0;\n    for (int i = 1; i <= n; ++i) {\n        cin >> leaf[i];\n        k = max(k, leaf[i]);\n    }\n    if (n == 1) {\n        cout << \"YES\\n\";\n        return;\n    }\n    pair<int, int> res = dfs(1, -1);\n    if (ok && res.first == 1 && res.second == k) {\n        cout << \"YES\\n\";\n    } else {\n        cout << \"NO\\n\";\n    }\n}\n\n```\n\n[2244G - Yura and Deadlines](https://codeforces.com/contest/2244/problem/G \"Codeforces Round 1109 (Div. 3)\")\n\n**Tutorial**\n\nTutorial is loading...\n\n**Solution**\n\n```\n#include <bits/stdc++.h>\n\nusing namespace std;\nusing ll = long long;\n\nstruct Fenwick {\n    int n;\n    vector<ll> t;\n    \n    Fenwick(int n) : n(n), t(n + 1, 0ll) {}\n    \n    int f(int x) {\n        return x & -x;\n    }\n    \n    void upd(int pos, ll val) {\n        for (; pos <= n; pos += f(pos)) t[pos] = max(t[pos], val);\n    }\n    \n    ll get(int pos) {\n        ll ans = 0;\n        for (; pos > 0; pos -= f(pos)) ans = max(ans, t[pos]);\n        return ans;\n    }\n};\n\nvoid I_love_feblokas() {\n    int n;\n    cin >> n;\n    vector<ll> a(n + 1);\n    for (int i = 1; i <= n; ++i) {\n        cin >> a[i];\n    }\n    Fenwick t(n);\n    vector<vector<pair<int, ll>>> ev(n + 1);\n    ll ans = 0;\n    for (int i = 1; i <= n; ++i) {\n        for (auto& ev : ev[i]) {\n            t.upd(ev.first, ev.second);\n        }\n        int lim = i - a[i] - 1;\n        ll mx = 0;\n        if (lim > 0) {\n            mx = t.get(min(n, lim));\n        }\n        ll dp_i = a[i] + mx;\n        ans = max(ans, dp_i);\n        ll act = i + a[i] + 1;\n        if (act <= n) {\n            ev[act].push_back({i, dp_i});\n        }\n    }\n    cout << ans << \"\\n\";\n}\n\nint main() {\n    int t;\n    cin >> t ;\n    while (t--) {\n        I_love_feblokas();\n    }\n    return 0;\n}\n\n```\n\n![](//codeforces.com/codeforces.org/s/90960/images/icons/paperclip-16x16.png) Tutorial of [Codeforces Round 1109 (Div. 3)](https://codeforces.com/contest/2244) \n\n* [![Vote: I like it](//codeforces.com/codeforces.org/s/90960/images/actions/voteup-gray.png \"Vote: I like it\")](#)\n* +39\n* [![Vote: I do not like it](//codeforces.com/codeforces.org/s/90960/images/actions/votedown-gray.png \"Vote: I do not like it\")](#)\n\n* [![Author](//codeforces.com/codeforces.org/s/90960/images/blog/user_16x16.png \"Author\")](https://codeforces.com/profile/itz%5Fpabloo) [itz\\_pabloo ](https://codeforces.com/profile/itz%5Fpabloo)\n* ![Publication date](//codeforces.com/codeforces.org/s/90960/images/blog/date_16x16.png \"Publication date\") 3 days ago\n* [![Comments](//codeforces.com/codeforces.org/s/90960/images/blog/comments_16x16.png \"Comments\")](https://codeforces.com/blog/entry/155197#comments) [120 ](https://codeforces.com/blog/entry/155197#comments)\n\n  \n![Comments](//codeforces.com/codeforces.org/s/90960/images/icons/comments-48x48.png \"Comments\") Comments (120) \n\n[Write comment?](#) \n\n| » [twinkelle](https://codeforces.com/profile/twinkelle \"Expert twinkelle\") | 3 days ago, [show (+1)](#) # |\n| -------------------------------------------------------------------------- | ---------------------------- |\n\n| » [ ![](https://codeforces.com/userpic.codeforces.org/5510515/avatar/98b79fff92760822.jpg) ](https://codeforces.com/profile/twinkelle) [twinkelle](https://codeforces.com/profile/twinkelle \"Expert twinkelle\") | 3 days ago, [hide](#) [#](https://codeforces.com/blog/entry/155197?#comment-1377727 \"Link to comment\") \\| [←](#)Rev. 2 [→](#) [![Vote: I like it](//codeforces.com/codeforces.org/s/90960/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) +7 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/90960/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) The solutions for B, E, F and G are not formatted properly [→](#) [Reply](#) |\n| --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n* | » » [itz\\_pabloo](https://codeforces.com/profile/itz%5Fpabloo \"Expert itz_pabloo\") | 3 days ago, [show (+1)](#) # |\n| ---------------------------------------------------------------------------------- | ---------------------------- |\n\n| » » [ ![](https://codeforces.com/userpic.codeforces.org/3446581/avatar/a137563d8f37c773.jpg) ](https://codeforces.com/profile/itz%5Fpabloo) [itz\\_pabloo](https://codeforces.com/profile/itz%5Fpabloo \"Expert itz_pabloo\") | 3 days ago, [hide](#) [#](https://codeforces.com/blog/entry/155197?#comment-1377730 \"Link to comment\") [^](#comment-1377727 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/90960/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) +1 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/90960/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) Fixed [→](#) [Reply](#) |\n| -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n  * | » » » [ShulkerBox](https://codeforces.com/profile/ShulkerBox \"Candidate Master ShulkerBox\") | 3 days ago, [show (+1)](#) # |\n| ------------------------------------------------------------------------------------------- | ---------------------------- |\n\n| » » » [ ![](https://codeforces.com/userpic.codeforces.org/5945276/avatar/ed21cb35f0671d81.jpg) ](https://codeforces.com/profile/ShulkerBox) [ShulkerBox](https://codeforces.com/profile/ShulkerBox \"Candidate Master ShulkerBox\") | 3 days ago, [hide](#) [#](https://codeforces.com/blog/entry/155197?#comment-1377745 \"Link to comment\") [^](#comment-1377730 \"Parent comment\") \\| [←](#)Rev. 2 [→](#) [![Vote: I like it](//codeforces.com/codeforces.org/s/90960/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) +1 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/90960/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) Am I missing some new C++ syntax updates? What is this line in the solution for $$$G$$$? int lim = i &mdash; a\\[i\\] &mdash; 1; [→](#) [Reply](#) |\n| --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n    * | » » » » [Angelator312](https://codeforces.com/profile/Angelator312 \"Pupil Angelator312\") | 3 days ago, [show (+1)](#) # |\n| ---------------------------------------------------------------------------------------- | ---------------------------- |\n\n| » » » » [ ![](https://codeforces.com/userpic.codeforces.org/4710575/avatar/c2a97dea464ac45.jpg) ](https://codeforces.com/profile/Angelator312) [Angelator312](https://codeforces.com/profile/Angelator312 \"Pupil Angelator312\") | 3 days ago, [hide](#) [#](https://codeforces.com/blog/entry/155197?#comment-1377766 \"Link to comment\") [^](#comment-1377745 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/90960/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/90960/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) If you see &mdash. Its just error in the showing the code or a typo made by the writer of the tutorial(not likely).These things are [HTML symbols](https://www.w3schools.com/html/html%5Fsymbols.asp) and have nothing to do with c++. [→](#) [Reply](#) |\n| ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n      * | » » » » » [KillGPT](https://codeforces.com/profile/KillGPT \"Newbie KillGPT\") | 3 days ago, [show](#) # |\n| ---------------------------------------------------------------------------- | ----------------------- |\n\n| » » » » » [ ![](https://codeforces.com/userpic.codeforces.org/4816799/avatar/85e9fb6de4f508fa.jpg) ](https://codeforces.com/profile/KillGPT) [KillGPT](https://codeforces.com/profile/KillGPT \"Newbie KillGPT\") | 3 days ago, [hide](#) [#](https://codeforces.com/blog/entry/155197?#comment-1377791 \"Link to comment\") [^](#comment-1377766 \"Parent comment\") \\| [←](#)Rev. 5 [→](#) [![Vote: I like it](//codeforces.com/codeforces.org/s/90960/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/90960/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) Yep, those are HTML entities not c++\"&mdash;\" means \"-\" minusyou can even see that in action on CF itself, just type &mdash; and you will see — after posting [→](#) [Reply](#) |\n| --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n| » [PiggyPump](https://codeforces.com/profile/PiggyPump \"Pupil PiggyPump\") | 3 days ago, [show (+1)](#) # |\n| ------------------------------------------------------------------------- | ---------------------------- |\n\n| » [ ![](https://codeforces.com/userpic.codeforces.org/no-avatar.jpg) ](https://codeforces.com/profile/PiggyPump) [PiggyPump](https://codeforces.com/profile/PiggyPump \"Pupil PiggyPump\") | 3 days ago, [hide](#) [#](https://codeforces.com/blog/entry/155197?#comment-1377731 \"Link to comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/90960/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/90960/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) Honestly a decent contest, I just haven't seen E before so I didn't know how to do it. [→](#) [Reply](#) |\n| ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n* | » » [BananamanCF](https://codeforces.com/profile/BananamanCF \"Pupil BananamanCF\") | 3 days ago, [show (+1)](#) # |\n| --------------------------------------------------------------------------------- | ---------------------------- |\n\n| » » [ ![](https://codeforces.com/userpic.codeforces.org/no-avatar.jpg) ](https://codeforces.com/profile/BananamanCF) [BananamanCF](https://codeforces.com/profile/BananamanCF \"Pupil BananamanCF\") | 3 days ago, [hide](#) [#](https://codeforces.com/blog/entry/155197?#comment-1377733 \"Link to comment\") [^](#comment-1377731 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/90960/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) +3 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/90960/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) you saw all the other problems before? [→](#) [Reply](#) |\n| -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n  * | » » » [PiggyPump](https://codeforces.com/profile/PiggyPump \"Pupil PiggyPump\") | 3 days ago, [show (+1)](#) # |\n| ----------------------------------------------------------------------------- | ---------------------------- |\n\n| » » » [ ![](https://codeforces.com/userpic.codeforces.org/no-avatar.jpg) ](https://codeforces.com/profile/PiggyPump) [PiggyPump](https://codeforces.com/profile/PiggyPump \"Pupil PiggyPump\") | 3 days ago, [hide](#) [#](https://codeforces.com/blog/entry/155197?#comment-1377738 \"Link to comment\") [^](#comment-1377733 \"Parent comment\") \\| [←](#)Rev. 3 [→](#) [![Vote: I like it](//codeforces.com/codeforces.org/s/90960/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) +9 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/90960/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) No I didn't \"see\" them before, but I kind of have.A — find longest continuous sequence, done that before.B — greedily place the books in a consecutive sequence 1,2,3... I have seen similar greedy problems where you have to place everybody consecutively.C — fun little graph problem — there was a similar usaco problem where you had to check if a cow and the position of the cow were in the same connected component.D — I listed out the array and the array times -1, realized you had to select the intervals according to the posts + Pref sums (aka DP) to speed up the solution — this problem was similar to We Be Flipping (Problem C from Spectral Cup Round 2)E — somebody help me here, I hate flipping subarrays in order to get it to alternate, I've never done this alternating shit before. It's obvious, but if you've never seen it — it's not, and I wasn't bothered enough to actually bash out the greedy approach? [→](#) [Reply](#) |\n| -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n    * | » » » » [mgranger22](https://codeforces.com/profile/mgranger22 \"Candidate Master mgranger22\") | 3 days ago, [show (+1)](#) # |\n| --------------------------------------------------------------------------------------------- | ---------------------------- |\n\n| » » » » [ ![](https://codeforces.com/userpic.codeforces.org/3637499/avatar/c2e543d66773ecca.jpg) ](https://codeforces.com/profile/mgranger22) [mgranger22](https://codeforces.com/profile/mgranger22 \"Candidate Master mgranger22\") | 3 days ago, [hide](#) [#](https://codeforces.com/blog/entry/155197?#comment-1377752 \"Link to comment\") [^](#comment-1377738 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/90960/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) +9 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/90960/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) My approach to E:Define $$$eqCnt$$$ as the count of $$$s\\[i\\] == s\\[i + 1\\]$$$. Clearly a string is alternating iff $$$eqCnt = 0$$$.Now consider what happens when you invert the range $$$\\[l, r\\]$$$. The only change in $$$eqCnt$$$ will come from neighbours $$$(l - 1, l)$$$ and $$$(r, r + 1)$$$, where one part is flipped, and the other is left alone. The inside of the region $$$\\[l, r\\]$$$, will have flips like $$$00$$$ to $$$11$$$ or $$$01$$$ to $$$10$$$, none of which will affect $$$eqCnt$$$.An optimal approach to make a string alternating is to iteratively fix the leftmost and rightmost occurences of equal neighbors. Suppose $$$s\\[l\\] = s\\[l + 1\\]$$$ and $$$s\\[r\\] = s\\[r + 1\\]$$$ then inverting $$$s\\[l : r\\]$$$ will reduce $$$eqCnt$$$ by $$$2$$$.A case to consider is when you only have $$$eqCnt = 1$$$. i.e. $$$01001$$$. In this case (given $$$s\\[i\\] == s\\[i + 1\\]$$$) you can just invert $$$s\\[i + 1 : n\\]$$$ and reach $$$eqCnt = 0$$$.This process can clear $$$eqCnt$$$ equal neighbors using $$$ \\\\lceil \\\\frac{equalCnt}{2} \\\\rceil$$$ operations.Thus for a $$$query(l, r, k)$$$, use prefix sums to get the $$$eqCnt$$$ of $$$s\\[l : r\\]$$$ and check if $$$\\\\lceil \\\\frac{equalCnt}{2} \\\\rceil \\\\leq k$$$. [→](#) [Reply](#) |\n| ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n      * | » » » » » [PiggyPump](https://codeforces.com/profile/PiggyPump \"Pupil PiggyPump\") | 3 days ago, [show](#) # |\n| --------------------------------------------------------------------------------- | ----------------------- |\n\n| » » » » » [ ![](https://codeforces.com/userpic.codeforces.org/no-avatar.jpg) ](https://codeforces.com/profile/PiggyPump) [PiggyPump](https://codeforces.com/profile/PiggyPump \"Pupil PiggyPump\") | 3 days ago, [hide](#) [#](https://codeforces.com/blog/entry/155197?#comment-1377764 \"Link to comment\") [^](#comment-1377752 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/90960/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/90960/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) Thank you mgranger22 — great explanation, way better than the editorial's by a mile! I didn't think about counting equalities, I was too distracted by counting alternations. It makes sense that the eqCnt of the interval from \\[l, r\\] always stays the same because you invert everything, some if 2 adj elements were equal they still are equal and vice versa.The only thing that changes the eqCnt is the relationship between l — 1 and l and r and r + 1, since l got flipped and r got flipped the eqCnt gets reduced by 2\\. Wow!And you can also reduce the last one (if eqCnt was originally odd) easily. Dang, well played. Now I know to look for equalities when you have alternating problem. [→](#) [Reply](#) |\n| ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n      * | » » » » » [me230003016](https://codeforces.com/profile/me230003016 \"Newbie me230003016\") | 3 days ago, [show (+1)](#) # |\n| ---------------------------------------------------------------------------------------- | ---------------------------- |\n\n| » » » » » [ ![](https://codeforces.com/userpic.codeforces.org/3447626/avatar/12c466e591586cf7.jpg) ](https://codeforces.com/profile/me230003016) [me230003016](https://codeforces.com/profile/me230003016 \"Newbie me230003016\") | 3 days ago, [hide](#) [#](https://codeforces.com/blog/entry/155197?#comment-1377767 \"Link to comment\") [^](#comment-1377752 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/90960/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/90960/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) I have a doubt in Question E, I was hoping you could help me,If k is equal to 0 and the string is not already alternating then shouldn't the answer automatically be NO [→](#) [Reply](#) |\n| ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n        * | » » » » » » [PiggyPump](https://codeforces.com/profile/PiggyPump \"Pupil PiggyPump\") | 3 days ago, [show](#) # |\n| ----------------------------------------------------------------------------------- | ----------------------- |\n\n| » » » » » » [ ![](https://codeforces.com/userpic.codeforces.org/no-avatar.jpg) ](https://codeforces.com/profile/PiggyPump) [PiggyPump](https://codeforces.com/profile/PiggyPump \"Pupil PiggyPump\") | 3 days ago, [hide](#) [#](https://codeforces.com/blog/entry/155197?#comment-1377770 \"Link to comment\") [^](#comment-1377767 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/90960/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/90960/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) Yes, I think so. [→](#) [Reply](#) |\n| -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n        * | » » » » » » [mgranger22](https://codeforces.com/profile/mgranger22 \"Candidate Master mgranger22\") | 3 days ago, [show](#) # |\n| ------------------------------------------------------------------------------------------------- | ----------------------- |\n\n| » » » » » » [ ![](https://codeforces.com/userpic.codeforces.org/3637499/avatar/c2e543d66773ecca.jpg) ](https://codeforces.com/profile/mgranger22) [mgranger22](https://codeforces.com/profile/mgranger22 \"Candidate Master mgranger22\") | 3 days ago, [hide](#) [#](https://codeforces.com/blog/entry/155197?#comment-1377773 \"Link to comment\") [^](#comment-1377767 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/90960/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/90960/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) Yes. This will be handled correctly by the above approach as $$$eqCnt \\\\gt 0$$$ implies $$$ \\\\lceil \\\\frac{eqCnt}{2} \\\\rceil \\\\gt 0$$$ which is not less than or equal to $$$k = 0$$$ [→](#) [Reply](#) |\n| --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ |\n        * | » » » » » » [me230003016](https://codeforces.com/profile/me230003016 \"Newbie me230003016\") | 3 days ago, [show](#) # |\n| ------------------------------------------------------------------------------------------ | ----------------------- |\n\n\n...233370 bytes truncated...\n\nThe tool call succeeded but the output was truncated. 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          "output": 1814,
          "reasoning": 3438,
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        },
        "modelID": "gpt-5.6-sol",
        "providerID": "openai",
        "time": {
          "created": 1784319834995,
          "completed": 1784319931696
        },
        "finish": "tool-calls",
        "structured": {
          "status": "success",
          "reason": null,
          "hints": [
            {
              "order": 1,
              "content": "Forget the actual bulb values for a moment. For every boundary between positions $i$ and $i+1$, mark it **bad** when $s_i=s_{i+1}$. A queried segment is beautiful exactly when it contains no bad boundaries."
            },
            {
              "order": 2,
              "content": "Study what flipping one interval does to these boundaries. If both bulbs of an adjacent pair are flipped, their equality relation does not change; the same is true if neither is flipped. Only boundaries crossed by an endpoint of the flipped interval can change."
            },
            {
              "order": 3,
              "content": "A flipped interval has at most two endpoint boundaries, so one operation can remove at most two bad boundaries. Therefore, if a query contains $c$ bad boundaries, at least $\\left\\lceil c/2\\right\\rceil$ operations are necessary."
            },
            {
              "order": 4,
              "content": "That lower bound is achievable. List the bad boundaries as $p_1<p_2<\\dots<p_c$. For each pair $p_{2j-1},p_{2j}$, flip $[p_{2j-1}+1,p_{2j}]$; this fixes exactly those two boundaries. If one boundary $p_c$ remains, flip $[p_c+1,r]$. Thus the exact minimum is $\\left\\lceil c/2\\right\\rceil$."
            },
            {
              "order": 5,
              "content": "Build a prefix sum over $a_i=[s_i=s_{i+1}]$. For query $[l,r]$, the number of bad boundaries is $c=P[r-1]-P[l-1]$, so print `YES` exactly when $(c+1)/2\\le k$. Each query is now $O(1)$; no segment-tree circus is required."
            }
          ],
          "editorial": "The whole problem collapses to one fact: if a queried segment contains $c$ equal adjacent pairs, then its exact minimum number of operations is\n\n$$\n\\left\\lceil \\frac{c}{2}\\right\\rceil.\n$$\n\nOnce we prove that, static prefix sums handle every query.\n\n## Bad boundaries\n\nCall the boundary between bulbs $i$ and $i+1$ **bad** if\n\n$$\ns_i=s_{i+1}.\n$$\n\nA binary segment is beautiful if and only if none of its internal boundaries are bad. Indeed, once every adjacent pair differs, the segment must be one of the two alternating patterns, depending only on its first bit.\n\nFor a query $[l,r]$, let\n\n$$\nc=\\#\\{i\\mid l\\le i<r,\\ s_i=s_{i+1}\\}.\n$$\n\nSo our job is simply to turn all $c$ bad boundaries into good ones.\n\n## What one flip can change\n\nSuppose we flip a subsegment $[x,y]$. Consider any adjacent pair:\n\n- If both bulbs are flipped, equality is preserved: `00` becomes `11`, `11` becomes `00`, and unequal pairs stay unequal.\n- If neither bulb is flipped, obviously nothing changes.\n- If exactly one bulb is flipped, equality reverses: a bad boundary becomes good, or a good boundary becomes bad.\n\nOnly the boundaries immediately outside the flipped subsegment, between $x-1,x$ and between $y,y+1$, have exactly one endpoint flipped. Therefore one operation changes at most two boundary states inside the queried segment.\n\nOperations extending outside $[l,r]$ offer no secret loophole: intersecting such an operation with $[l,r]$ flips exactly the same queried bulbs. We may therefore assume every useful operation lies inside the query.\n\nSince one operation can fix at most two bad boundaries, any solution needs at least\n\n$$\n\\left\\lceil \\frac{c}{2}\\right\\rceil\n$$\n\noperations.\n\n## The lower bound is attainable\n\nList the bad boundaries in increasing order:\n\n$$\np_1<p_2<\\dots<p_c,\n$$\n\nwhere boundary $p_j$ lies between bulbs $p_j$ and $p_j+1$.\n\nTake two consecutive bad boundaries $p_{2j-1}$ and $p_{2j}$ and flip\n\n$$\n[p_{2j-1}+1,\\ p_{2j}].\n$$\n\nAt boundary $p_{2j-1}$, only the right bulb is flipped. At boundary $p_{2j}$, only the left bulb is flipped. Thus both bad boundaries become good. Every boundary strictly between them has both bulbs flipped, so its state is unchanged. No other internal boundary is touched.\n\nThis fixes each pair of bad boundaries with one operation. If $c$ is odd, one bad boundary $p_c$ remains; flipping\n\n$$\n[p_c+1,r]\n$$\n\nchanges that boundary and no other internal boundary of the query.\n\nHence we can always succeed in\n\n$$\n\\left\\lfloor\\frac c2\\right\\rfloor+(c\\bmod 2)\n=\\left\\lceil\\frac c2\\right\\rceil\n$$\n\noperations. Combined with the lower bound, this is the exact optimum. The interval-flip drama was really just endpoint bookkeeping.\n\n## Answering all queries\n\nDefine\n\n$$\na_i=[s_i=s_{i+1}]\\qquad(1\\le i<n)\n$$\n\nand prefix sums\n\n$$\nP[0]=0,\\qquad P[i]=\\sum_{j=1}^{i}a_j.\n$$\n\nThe bad boundaries internal to $[l,r]$ have indices $l,l+1,\\dots,r-1$, so\n\n$$\nc=P[r-1]-P[l-1].\n$$\n\nThe answer is therefore `YES` exactly when\n\n$$\n\\frac{c+1}{2}\\le k,\n$$\n\nwhere the division is integer division. This also handles $l=r$: then $c=0$, and a one-bulb segment is already beautiful.\n\n## Correctness proof\n\nFor every query $[l,r]$:\n\n1. A segment is beautiful exactly when all its internal bad boundaries have been removed.\n2. Any interval flip changes at most two internal boundary states, so removing $c$ bad boundaries requires at least $\\lceil c/2\\rceil$ operations.\n3. Pairing bad boundaries and flipping between each pair removes two per operation without changing any others; an odd leftover is removed by flipping a suffix of the query.\n4. Therefore the minimum required number of operations is exactly $\\lceil c/2\\rceil$, which the algorithm compares with $k$.\n\nThus every printed answer is correct.\n\n## Complexity\n\nFor each test case, preprocessing takes $O(n)$ time. Every query takes $O(1)$ time, for total time $O(n+q)$ and memory $O(n)$.",
          "solution": "#include <bits/stdc++.h>\nusing namespace std;\n\nusing ll = long long;\n\nvoid setIO() {\n    ios::sync_with_stdio(false);\n    cin.tie(nullptr);\n}\n\nint main() {\n    setIO();\n\n    int t;\n    cin >> t;\n    while (t--) {\n        int n, q;\n        cin >> n >> q;\n\n        string s;\n        cin >> s;\n\n        vector<int> pref(n);\n        for (int i = 1; i < n; ++i) {\n            pref[i] = pref[i - 1] + (s[i - 1] == s[i]);\n        }\n\n        while (q--) {\n            int l, r, k;\n            cin >> l >> r >> k;\n\n            int bad = pref[r - 1] - pref[l - 1];\n            cout << ((bad + 1) / 2 <= k ? \"YES\\n\" : \"NO\\n\");\n        }\n    }\n\n    return 0;\n}"
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      "parts": [
        {
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          "messageID": "msg_f71bffb73001E1EDwW7SLgCDNs"
        },
        {
          "type": "reasoning",
          "text": "**Verifying operation constraints and formula assumptions**",
          "time": {
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            "end": 1784319845186
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          "messageID": "msg_f71bffb73001E1EDwW7SLgCDNs"
        },
        {
          "type": "reasoning",
          "text": "**Deriving operation effect on adjacent pairs**",
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          "messageID": "msg_f71bffb73001E1EDwW7SLgCDNs"
        },
        {
          "type": "reasoning",
          "text": "**Reframing problem using mismatch vectors**",
          "time": {
            "start": 1784319847136,
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          "type": "reasoning",
          "text": "**Analyzing run counts and parity transformations**",
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        {
          "type": "reasoning",
          "text": "**Designing prefix array for boundary queries**",
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          "tool": "StructuredOutput",
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              "status": "success",
              "reason": null,
              "hints": [
                {
                  "order": 1,
                  "content": "Forget the actual bulb values for a moment. For every boundary between positions $i$ and $i+1$, mark it **bad** when $s_i=s_{i+1}$. A queried segment is beautiful exactly when it contains no bad boundaries."
                },
                {
                  "order": 2,
                  "content": "Study what flipping one interval does to these boundaries. If both bulbs of an adjacent pair are flipped, their equality relation does not change; the same is true if neither is flipped. Only boundaries crossed by an endpoint of the flipped interval can change."
                },
                {
                  "order": 3,
                  "content": "A flipped interval has at most two endpoint boundaries, so one operation can remove at most two bad boundaries. Therefore, if a query contains $c$ bad boundaries, at least $\\left\\lceil c/2\\right\\rceil$ operations are necessary."
                },
                {
                  "order": 4,
                  "content": "That lower bound is achievable. List the bad boundaries as $p_1<p_2<\\dots<p_c$. For each pair $p_{2j-1},p_{2j}$, flip $[p_{2j-1}+1,p_{2j}]$; this fixes exactly those two boundaries. If one boundary $p_c$ remains, flip $[p_c+1,r]$. Thus the exact minimum is $\\left\\lceil c/2\\right\\rceil$."
                },
                {
                  "order": 5,
                  "content": "Build a prefix sum over $a_i=[s_i=s_{i+1}]$. For query $[l,r]$, the number of bad boundaries is $c=P[r-1]-P[l-1]$, so print `YES` exactly when $(c+1)/2\\le k$. Each query is now $O(1)$; no segment-tree circus is required."
                }
              ],
              "editorial": "The whole problem collapses to one fact: if a queried segment contains $c$ equal adjacent pairs, then its exact minimum number of operations is\n\n$$\n\\left\\lceil \\frac{c}{2}\\right\\rceil.\n$$\n\nOnce we prove that, static prefix sums handle every query.\n\n## Bad boundaries\n\nCall the boundary between bulbs $i$ and $i+1$ **bad** if\n\n$$\ns_i=s_{i+1}.\n$$\n\nA binary segment is beautiful if and only if none of its internal boundaries are bad. Indeed, once every adjacent pair differs, the segment must be one of the two alternating patterns, depending only on its first bit.\n\nFor a query $[l,r]$, let\n\n$$\nc=\\#\\{i\\mid l\\le i<r,\\ s_i=s_{i+1}\\}.\n$$\n\nSo our job is simply to turn all $c$ bad boundaries into good ones.\n\n## What one flip can change\n\nSuppose we flip a subsegment $[x,y]$. Consider any adjacent pair:\n\n- If both bulbs are flipped, equality is preserved: `00` becomes `11`, `11` becomes `00`, and unequal pairs stay unequal.\n- If neither bulb is flipped, obviously nothing changes.\n- If exactly one bulb is flipped, equality reverses: a bad boundary becomes good, or a good boundary becomes bad.\n\nOnly the boundaries immediately outside the flipped subsegment, between $x-1,x$ and between $y,y+1$, have exactly one endpoint flipped. Therefore one operation changes at most two boundary states inside the queried segment.\n\nOperations extending outside $[l,r]$ offer no secret loophole: intersecting such an operation with $[l,r]$ flips exactly the same queried bulbs. We may therefore assume every useful operation lies inside the query.\n\nSince one operation can fix at most two bad boundaries, any solution needs at least\n\n$$\n\\left\\lceil \\frac{c}{2}\\right\\rceil\n$$\n\noperations.\n\n## The lower bound is attainable\n\nList the bad boundaries in increasing order:\n\n$$\np_1<p_2<\\dots<p_c,\n$$\n\nwhere boundary $p_j$ lies between bulbs $p_j$ and $p_j+1$.\n\nTake two consecutive bad boundaries $p_{2j-1}$ and $p_{2j}$ and flip\n\n$$\n[p_{2j-1}+1,\\ p_{2j}].\n$$\n\nAt boundary $p_{2j-1}$, only the right bulb is flipped. At boundary $p_{2j}$, only the left bulb is flipped. Thus both bad boundaries become good. Every boundary strictly between them has both bulbs flipped, so its state is unchanged. No other internal boundary is touched.\n\nThis fixes each pair of bad boundaries with one operation. If $c$ is odd, one bad boundary $p_c$ remains; flipping\n\n$$\n[p_c+1,r]\n$$\n\nchanges that boundary and no other internal boundary of the query.\n\nHence we can always succeed in\n\n$$\n\\left\\lfloor\\frac c2\\right\\rfloor+(c\\bmod 2)\n=\\left\\lceil\\frac c2\\right\\rceil\n$$\n\noperations. Combined with the lower bound, this is the exact optimum. The interval-flip drama was really just endpoint bookkeeping.\n\n## Answering all queries\n\nDefine\n\n$$\na_i=[s_i=s_{i+1}]\\qquad(1\\le i<n)\n$$\n\nand prefix sums\n\n$$\nP[0]=0,\\qquad P[i]=\\sum_{j=1}^{i}a_j.\n$$\n\nThe bad boundaries internal to $[l,r]$ have indices $l,l+1,\\dots,r-1$, so\n\n$$\nc=P[r-1]-P[l-1].\n$$\n\nThe answer is therefore `YES` exactly when\n\n$$\n\\frac{c+1}{2}\\le k,\n$$\n\nwhere the division is integer division. This also handles $l=r$: then $c=0$, and a one-bulb segment is already beautiful.\n\n## Correctness proof\n\nFor every query $[l,r]$:\n\n1. A segment is beautiful exactly when all its internal bad boundaries have been removed.\n2. Any interval flip changes at most two internal boundary states, so removing $c$ bad boundaries requires at least $\\lceil c/2\\rceil$ operations.\n3. Pairing bad boundaries and flipping between each pair removes two per operation without changing any others; an odd leftover is removed by flipping a suffix of the query.\n4. Therefore the minimum required number of operations is exactly $\\lceil c/2\\rceil$, which the algorithm compares with $k$.\n\nThus every printed answer is correct.\n\n## Complexity\n\nFor each test case, preprocessing takes $O(n)$ time. Every query takes $O(1)$ time, for total time $O(n+q)$ and memory $O(n)$.",
              "solution": "#include <bits/stdc++.h>\nusing namespace std;\n\nusing ll = long long;\n\nvoid setIO() {\n    ios::sync_with_stdio(false);\n    cin.tie(nullptr);\n}\n\nint main() {\n    setIO();\n\n    int t;\n    cin >> t;\n    while (t--) {\n        int n, q;\n        cin >> n >> q;\n\n        string s;\n        cin >> s;\n\n        vector<int> pref(n);\n        for (int i = 1; i < n; ++i) {\n            pref[i] = pref[i - 1] + (s[i - 1] == s[i]);\n        }\n\n        while (q--) {\n            int l, r, k;\n            cin >> l >> r >> k;\n\n            int bad = pref[r - 1] - pref[l - 1];\n            cout << ((bad + 1) / 2 <= k ? \"YES\\n\" : \"NO\\n\");\n        }\n    }\n\n    return 0;\n}"
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            "output": "Structured output captured successfully.",
            "metadata": {
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            "title": "Structured Output",
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