<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 dp[10][10][10][10][140];
    
    int fn(int mx, int my, int cx, int cy, int moves, vector<string>& grid, int catJump, int mouseJump) {
        int n = grid.size();
        int m = grid[0].size();
        
        if (moves >= 128) return 2;  // Avoid infinite loops
        if (grid[mx][my] == 'F') return 1;  // Mouse wins
        if (grid[cx][cy] == 'F') return 2;  // Cat wins
        if (mx == cx && my == cy) return 2;  // Cat catches mouse
        
        if (dp[mx][my][cx][cy][moves] != -1) return dp[mx][my][cx][cy][moves];

        int del_x[5] = {0, -1, 0, 1,0};
        int del_y[5] = {-1, 0, 1, 0,0};

        if (moves % 2 == 0) {  // Mouse's turn
            for (int i = 0; i < 5; i++) {
                for (int p = 1; p <= mouseJump; p++) {
                    int nx = mx + del_x[i] * p;
                    int ny = my + del_y[i] * p;

                    if (nx >= 0 && nx < n && ny >= 0 && ny < m) {
                        if (grid[nx][ny] == '#') break;
                        int outcome = fn(nx, ny, cx, cy, moves + 1, grid, catJump, mouseJump);
                        if (outcome == 1) return dp[mx][my][cx][cy][moves] = 1;  // Mouse wins
                    }
                }
            }
            return dp[mx][my][cx][cy][moves] = 2;  // Cat wins if no valid move
        } else {  // Cat's turn
            for (int i = 0; i < 5; i++) {
                for (int p = 1; p <= catJump; p++) {
                    int nx = cx + del_x[i] * p;
                    int ny = cy + del_y[i] * p;

                    if (nx >= 0 && nx < n && ny >= 0 && ny < m) {
                        if (grid[nx][ny] == '#') break;
                        int outcome = fn(mx, my, nx, ny, moves + 1, grid, catJump, mouseJump);
                        if (outcome == 2) return dp[mx][my][cx][cy][moves] = 2;  // Cat wins
                    } 
                }
            }
            return dp[mx][my][cx][cy][moves] = 1;  // Mouse wins if no valid move
        }
    }

    bool canMouseWin(vector<string>& grid, int catJump, int mouseJump) {
        memset(dp, -1, sizeof(dp));
        int mx, my, cx, cy;
        
        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;
                }
                if (grid[i][j] == 'M') {
                    mx = i;
                    my = j; 
                }
            }
        }

        int ans = fn(mx, my, cx, cy, 0, grid, catJump, mouseJump);
        return ans == 1;
    }
};

17900