<p>A game is played by a cat and a mouse named Cat and Mouse.</p>

<p>The environment is represented by a <code>grid</code> of size <code>rows x cols</code>, where each element is a wall, floor, player (Cat, Mouse), or food.</p>

<ul>
	<li>Players are represented by the characters <code>&#39;C&#39;</code>(Cat)<code>,&#39;M&#39;</code>(Mouse).</li>
	<li>Floors are represented by the character <code>&#39;.&#39;</code> and can be walked on.</li>
	<li>Walls are represented by the character <code>&#39;#&#39;</code> and cannot be walked on.</li>
	<li>Food is represented by the character <code>&#39;F&#39;</code> and can be walked on.</li>
	<li>There is only one of each character <code>&#39;C&#39;</code>, <code>&#39;M&#39;</code>, and <code>&#39;F&#39;</code> in <code>grid</code>.</li>
</ul>

<p>Mouse and Cat play according to the following rules:</p>

<ul>
	<li>Mouse <strong>moves first</strong>, then they take turns to move.</li>
	<li>During each turn, Cat and Mouse can jump in one of the four directions (left, right, up, down). They cannot jump over the wall nor outside of the <code>grid</code>.</li>
	<li><code>catJump, mouseJump</code> are the maximum lengths Cat and Mouse can jump at a time, respectively. Cat and Mouse can jump less than the maximum length.</li>
	<li>Staying in the same position is allowed.</li>
	<li>Mouse can jump over Cat.</li>
</ul>

<p>The game can end in 4 ways:</p>

<ul>
	<li>If Cat occupies the same position as Mouse, Cat wins.</li>
	<li>If Cat reaches the food first, Cat wins.</li>
	<li>If Mouse reaches the food first, Mouse wins.</li>
	<li>If Mouse cannot get to the food within 1000 turns, Cat wins.</li>
</ul>

<p>Given a <code>rows x cols</code> matrix <code>grid</code> and two integers <code>catJump</code> and <code>mouseJump</code>, return <code>true</code><em> if Mouse can win the game if both Cat and Mouse play optimally, otherwise return </em><code>false</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/09/12/sample_111_1955.png" style="width: 580px; height: 239px;" />
<pre>
<strong>Input:</strong> grid = [&quot;####F&quot;,&quot;#C...&quot;,&quot;M....&quot;], catJump = 1, mouseJump = 2
<strong>Output:</strong> true
<strong>Explanation:</strong> Cat cannot catch Mouse on its turn nor can it get the food before Mouse.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/09/12/sample_2_1955.png" style="width: 580px; height: 175px;" />
<pre>
<strong>Input:</strong> grid = [&quot;M.C...F&quot;], catJump = 1, mouseJump = 4
<strong>Output:</strong> true
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> grid = [&quot;M.C...F&quot;], catJump = 1, mouseJump = 3
<strong>Output:</strong> false
</pre>

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

<ul>
	<li><code>rows == grid.length</code></li>
	<li><code>cols = grid[i].length</code></li>
	<li><code>1 &lt;= rows, cols &lt;= 8</code></li>
	<li><code>grid[i][j]</code> consist only of characters <code>&#39;C&#39;</code>, <code>&#39;M&#39;</code>, <code>&#39;F&#39;</code>, <code>&#39;.&#39;</code>, and <code>&#39;#&#39;</code>.</li>
	<li>There is only one of each character <code>&#39;C&#39;</code>, <code>&#39;M&#39;</code>, and <code>&#39;F&#39;</code> in <code>grid</code>.</li>
	<li><code>1 &lt;= catJump, mouseJump &lt;= 8</code></li>
</ul>

class Solution {
    vector<vector<int>> dirs = {{0, 1}, {0, -1}, {1, 0}, {-1, 0}};
    unordered_map<int, bool> memo;

    int getMemoKey(int turn, int mx, int my, int cx, int cy) {
        return (turn << 12) + (mx << 9) + (my << 6) + (cx << 3) + cy;
    }
    int maxTurn = 1000;

public:
    bool canMouseWin(vector<string>& grid, int catJump, int mouseJump) {
        int mx, my, cx, cy;
        int cnt = 0;

        for (int i = 0; i < grid.size(); ++i) {
            for (int j = 0; j < grid[0].size(); ++j) {
                if (grid[i][j] == 'C') {
                    cx = i;
                    cy = j;
                    grid[i][j] = '.';
                } else if (grid[i][j] == 'M') {
                    mx = i;
                    my = j;
                    grid[i][j] = '.';
                }

                cnt += grid[i][j] != '#';
            }
        }

        maxTurn = cnt*2 + 1;

        return checkCanWin(grid, mx, my, cx, cy, 0, catJump, mouseJump);
    }

    bool checkCanWin(vector<string>& grid, int mx, int my, int cx, int cy, int turn, int catJump, int mouseJump) {
        if (turn >= maxTurn) return false;
        if (cx == mx && cy == my) return false;

        int memoKey = getMemoKey(turn, mx, my, cx, cy);

        if (memo.contains(memoKey)) return memo[memoKey];

        int m = grid.size();
        int n = grid[0].size();
        int win = checkCanWin(grid, mx, my, cx, cy, turn+1, catJump, mouseJump);

        if (turn % 2 == 0) {
            if (win) {
                return memo[memoKey] = true;
            }
            for (auto &dir : dirs) {
                for (int j = 1; j <= mouseJump; ++j) {
                    int x = mx + dir[0]*j;
                    int y = my + dir[1]*j;

                    if (x < 0 || x >= m || y < 0 || y >= n || grid[x][y] == '#') break;

                    if (grid[x][y] == 'F' || checkCanWin(grid, x, y, cx, cy, turn+1, catJump, mouseJump)) {
                        return memo[memoKey] = true;
                    }
                }
            }

            return memo[memoKey] = false;
        }

        if (!win) {
            return memo[memoKey] = false;
        }

        for (auto &dir : dirs) {
            for (int j = 1; j <= catJump; ++j) {
                int x = cx + dir[0]*j;
                int y = cy + dir[1]*j;

                if (x < 0 || x >= m || y < 0 || y >= n || grid[x][y] == '#') break;

                if (grid[x][y] == 'F' || !checkCanWin(grid, mx, my, x, y, turn+1, catJump, mouseJump)) {
                    return memo[memoKey] = false;
                }
            }
        }

        return memo[memoKey] = true;
    }
};
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