<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:
    def canMouseWin(self, grid: List[str], catJump: int, mouseJump: int) -> bool:
        dp = {}

        dirs = [[0, 1], [0, -1], [1, 0], [-1, 0]]

        def in_bound(pos):
            return pos[0] >= 0 and pos[1] >= 0 and pos[0] < len(grid) and pos[1] < len(grid[0])
        
        def dfs(c_pos, m_pos, turn):
            # print(c_pos, m_pos, turn)
            nonlocal dp

            if c_pos == m_pos:
                dp[(c_pos, m_pos, turn)] = False
                return False
            
            if grid[c_pos[0]][c_pos[1]] == 'F':
                dp[(c_pos, m_pos, turn)] = False
                return False

            if grid[m_pos[0]][m_pos[1]] == 'F':
                dp[(c_pos, m_pos, turn)] = True
                return True
            
            if turn > 100:
                dp[(c_pos, m_pos, turn)] = False
                return False
            
            if((c_pos, m_pos, turn) in dp.keys()):
                return dp[(c_pos, m_pos, turn)]
            
            if turn % 2 == 0:
                ans=False
                for d in dirs:
                    for k in range(1, mouseJump+1):
                        ne = (m_pos[0] + k * d[0], m_pos[1] + k* d[1])
                        if in_bound(ne) and grid[ne[0]][ne[1]] != '#':
                            ans = ans or dfs(c_pos, ne, turn+1)
                        else:
                            break
                dp[(c_pos, m_pos, turn)] = ans
                return ans
            else:
                ans=True
                for d in dirs:
                    for k in range(catJump+1):
                        ne = (c_pos[0] + k * d[0], c_pos[1] + k* d[1])
                        if in_bound(ne) and grid[ne[0]][ne[1]] != '#':
                            ans = ans and dfs(ne, m_pos, turn+1)
                        else:
                            break
                dp[(c_pos, m_pos, turn)] = ans
                return ans
        
        c_pos = None
        m_pos = None
        for i in range(len(grid)):
            for j in range(len(grid[0])):
                if(grid[i][j] == 'C'):
                    c_pos = (i, j)
                if(grid[i][j] == 'M'):
                    m_pos = (i,j)
                if(c_pos != None and m_pos != None):
                    break
        
        return dfs(c_pos, m_pos, 0)

                        
                        

        
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