<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[9][9][9][9][140];
    int dx[4] = {0,1,0,-1};
    int dy[4] = {1,0,-1,0};
    
    bool solve(vector<string>& grid, int catJump, int mouseJump, int catR, int catC, int mouseR, int mouseC, int moves){        
        int r = grid.size(), c = grid[0].size();
        if(moves >= 128) return false;
        if(mouseR==catR && mouseC==catC) return false;
        if(grid[catR][catC] == 'F') return false;
        if(grid[mouseR][mouseC] == 'F') return true;
        
        if(dp[catR][catC][mouseR][mouseC][moves] != -1) 
            return dp[catR][catC][mouseR][mouseC][moves];
        
        if(moves%2 == 0){
            for(int i=0; i<4; ++i){
                for(int j=0; j<=mouseJump; ++j){
                    int x = mouseR+j*dx[i], y = mouseC+j*dy[i];
                    if(x>=0 && x<r && y>=0 && y<c && grid[x][y]!='#'){
                        if(solve(grid, catJump, mouseJump, catR, catC, x, y, moves+1) == true) 
                            return dp[catR][catC][mouseR][mouseC][moves] = true;
                    }else break;
                }
            }
            return dp[catR][catC][mouseR][mouseC][moves] = false;
        }else{
            for(int i=0; i<4; ++i){
                for(int j=0; j<=catJump; ++j){
                    int x = catR+dx[i]*j, y = catC+dy[i]*j;
                    if(x>=0 && x<r && y>=0 && y<c && grid[x][y]!='#'){
                        if(solve(grid, catJump, mouseJump, x, y, mouseR, mouseC, moves+1) == false) 
                            return dp[catR][catC][mouseR][mouseC][moves] = false; 
                    }else break;
                }
            }
            return dp[catR][catC][mouseR][mouseC][moves] = true;
        }
    }
    
    bool canMouseWin(vector<string>& grid, int catJump, int mouseJump) {
       memset(dp,-1,sizeof(dp));
        int r = grid.size(), c =grid[0].size();
        int catR, catC, mouseR, mouseC;
        for(int i=0; i<r; ++i){
            for(int j=0; j<c; ++j){
                if(grid[i][j] == 'C'){
                    catR = i, catC = j;
                }
                else if(grid[i][j] == 'M'){
                    mouseR = i, mouseC = j;
                }
            }
        }
        
        return solve(grid, catJump, mouseJump, catR, catC, mouseR, mouseC, 0);
    }
};
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