<p>Alice and Bob continue their games with piles of stones. There are a number of piles <strong>arranged in a row</strong>, and each pile has a positive integer number of stones <code>piles[i]</code>. The objective of the game is to end with the most stones.</p>

<p>Alice and Bob take turns, with Alice starting first.</p>

<p>On each player&#39;s turn, that player can take <strong>all the stones</strong> in the <strong>first</strong> <code>X</code> remaining piles, where <code>1 &lt;= X &lt;= 2M</code>. Then, we set <code>M = max(M, X)</code>. Initially, M = 1.</p>

<p>The game continues until all the stones have been taken.</p>

<p>Assuming Alice and Bob play optimally, return the maximum number of stones Alice can get.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">piles = [2,7,9,4,4]</span></p>

<p><strong>Output:</strong> <span class="example-io">10</span></p>

<p><strong>Explanation:</strong></p>

<ul>
	<li>If Alice takes one pile at the beginning, Bob takes two piles, then Alice takes 2 piles again. Alice can get <code>2 + 4 + 4 = 10</code> stones in total.</li>
	<li>If Alice takes two piles at the beginning, then Bob can take all three piles left. In this case, Alice get <code>2 + 7 = 9</code> stones in total.</li>
</ul>

<p>So we return 10 since it&#39;s larger.</p>
</div>

<p><strong class="example">Example 2:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">piles = [1,2,3,4,5,100]</span></p>

<p><strong>Output:</strong> <span class="example-io">104</span></p>
</div>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= piles.length &lt;= 100</code></li>
	<li><code>1 &lt;= piles[i]&nbsp;&lt;= 10<sup>4</sup></code></li>
</ul>

struct Game {
    int idx;
    int M;
    bool isAlice;

    bool operator==(const Game& other) const {
        return (idx == other.idx &&  M == other.M &&  isAlice == other.isAlice);
    }
};

struct Hash {
    std::size_t operator()(const Game& s) const {
        std::size_t h1 = std::hash<int>{}(s.idx);
        std::size_t h2 = std::hash<int>{}(s.M);
        std::size_t h3 = std::hash<bool>{}(s.isAlice);
        return h1 ^ (h2 << 1) ^ (h3 << 2);
    }
};

class Solution {
public:
    int stoneGameII(vector<int>& piles) {
        unordered_map<Game, pair<int, int>, Hash> cache;
        auto [alice, bob] = dp(piles, 0, true, 1, cache);
        return alice;
    }

    pair<int, int> dp(vector<int>& piles, int idx, bool isAliceTurn, int M, unordered_map<Game, pair<int, int>, Hash>& cache) {
        if (idx >= piles.size()) return {0, 0};
        if (cache.contains({idx, M, isAliceTurn})) return cache[{idx, M, isAliceTurn}];
        int maxAlice = 0, maxBob = 0;
        int alice = 0, bob = 0;
        for (int X = 1; (X <= 2 * M) && ((idx + X) <= piles.size()); ++X) {
            if (isAliceTurn) alice += piles[idx + X - 1];
            else bob += piles[idx + X - 1];
            auto [aliceTail, bobTail] = dp(piles, idx + X, !isAliceTurn, max(M, X), cache);
            bool isBetter = false;
            int aliceNew = alice + aliceTail, bobNew = bob + bobTail;
            if (isAliceTurn) {
                isBetter = (aliceNew > maxAlice) || (aliceNew == maxAlice && bobNew <= maxBob);
            } else {
                isBetter = (bobNew > maxBob) || (bobNew == maxBob && aliceNew <= maxAlice);
            }
            if (isBetter) {
                maxAlice = aliceNew;
                maxBob = bobNew;
            }
        }
        cache[{idx, M, isAliceTurn}] = {maxAlice, maxBob};
        return cache[{idx, M, isAliceTurn}];
    }
};
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