<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 {
public:
    int n, m;
    int dx[4] = {-1, 0, 0, 1};
    int dy[4] = {0, -1, 1, 0};
    int dp[1005][8][8][8][8];
    vector<int> cat, food, mouse;
    int solve(int c0, int c1, int m0, int m1, vector<string>& grid, int cj,
              int mj, int turn) {
        if (turn >= 128)
            return 0;

        if (dp[turn][m0][m1][c0][c1] != -1)
            return dp[turn][m0][m1][c0][c1];
        int ret = turn % 2 == 0 ? 0 : 1;
        if (turn % 2 == 0) // mouse
        {
            if (c0 == m0 && c1 == m1)
                ret = 0;
            else if (m0 == food[0] && m1 == food[1])
                ret = 1;
            else if (c0 == food[0] && c1 == food[1])
                ret = 0;
            else {
                for (int i = 0; i < 4; i++) {
                    for (int j = 0; j <= mj; j++) {
                        int x = m0 + j * dx[i];
                        int y = m1 + j * dy[i];

                        if (x >= 0 && y >= 0 && y < m && x < n &&
                            grid[x][y] != '#') {
                            if (solve(c0, c1, x, y, grid, cj, mj, turn + 1)) {
                                ret = 1;
                                break;
                            }
                        } else
                            break;
                    }
                }
            }

        } else {
            if (c0 == m0 && c1 == m1)
                ret = 0;
            else if (c0 == food[0] && c1 == food[1])
                ret = 0;
            else if (m0 == food[0] && m1 == food[1])
                ret = 1;
            else {
                for (int i = 0; i < 4; i++) {
                    for (int j = 0; j <= cj; j++) {
                        int x = c0 + j * dx[i];
                        int y = c1 + j * dy[i];

                        if (x >= 0 && y >= 0 && y < m && x < n &&
                            grid[x][y] != '#') {
                            if (solve(x, y, m0, m1, grid, cj, mj, turn + 1) ==
                                0) {
                                ret = 0;
                                break;
                            }
                        } else
                            break;
                    }
                }
            }
        }
        return dp[turn][m0][m1][c0][c1] = ret;
    }
    bool canMouseWin(vector<string>& grid, int catJump, int mouseJump) {
        memset(dp, -1, sizeof(dp));
        cat.resize(2);
        mouse.resize(2);
        food.resize(2);
        int R = grid.size();
        int C = grid[0].size();
        n = R;
        m = C;
        for (int i = 0; i < R; i++) {
            for (int j = 0; j < C; j++) {
                if (grid[i][j] == 'C')
                    cat[0] = i, cat[1] = j;
                else if (grid[i][j] == 'M')
                    mouse[0] = i, mouse[1] = j;
                else if (grid[i][j] == 'F')
                    food[0] = i, food[1] = j;
            }
        }
        return solve(cat[0], cat[1], mouse[0], mouse[1], grid, catJump,
                     mouseJump, 0);
    }
};
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