<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:
        rows = len(grid)
        cols = len(grid[0])
        totalCells = 0
        catPos = 0
        mousePos = 0
        for r in range(rows):
            for c in range(cols):
                if grid[r][c] != '#':
                    totalCells += 1
                if grid[r][c] == 'C':
                    catPos = self.encode(r, c, cols)
                elif grid[r][c] == 'M':
                    mousePos = self.encode(r, c, cols)
        memo = [[[None] * (totalCells * 2) for _ in range(rows * cols)] for _ in range(rows * cols)]
        return self.dfs(grid, catPos, mousePos, 0, catJump, mouseJump, rows, cols, totalCells, memo)

    def dfs(self, grid: list[str], catPos: int, mousePos: int, turn: int, catJump: int,
            mouseJump: int, rows: int, cols: int, totalCells: int, memo: list[list[list[bool]]]) -> bool:
        if turn == totalCells * 2:
            return False
        if memo[catPos][mousePos][turn] is not None:
            return memo[catPos][mousePos][turn]
        if turn % 2 == 0:
            r, c = divmod(mousePos, cols)
            for dx, dy in [(0, 1), (1, 0), (0, -1), (-1, 0)]:
                for j in range(mouseJump + 1):
                    nr, nc = r + dx * j, c + dy * j
                    if nr < 0 or nr >= rows or nc < 0 or nc >= cols:
                        break
                    if grid[nr][nc] == '#':
                        break
                    if grid[nr][nc] == 'F':
                        return self.setMemoAndReturn(memo, catPos, mousePos, turn, True)
                    if self.dfs(grid, catPos, self.encode(nr, nc, cols), turn + 1, catJump, mouseJump, rows, cols, totalCells, memo):
                        return self.setMemoAndReturn(memo, catPos, mousePos, turn, True)
            return self.setMemoAndReturn(memo, catPos, mousePos, turn, False)
        else:
            r, c = divmod(catPos, cols)
            for dx, dy in [(0, 1), (1, 0), (0, -1), (-1, 0)]:
                for j in range(catJump + 1):
                    nr, nc = r + dx * j, c + dy * j
                    if nr < 0 or nr >= rows or nc < 0 or nc >= cols:
                        break
                    if grid[nr][nc] == '#':
                        break
                    if grid[nr][nc] == 'F':
                        return self.setMemoAndReturn(memo, catPos, mousePos, turn, False)
                    nextCatPos = self.encode(nr, nc, cols)
                    if nextCatPos == mousePos:
                        return self.setMemoAndReturn(memo, catPos, mousePos, turn, False)
                    if not self.dfs(grid, nextCatPos, mousePos, turn + 1, catJump, mouseJump, rows, cols, totalCells, memo):
                        return self.setMemoAndReturn(memo, catPos, mousePos, turn, False)
            return self.setMemoAndReturn(memo, catPos, mousePos, turn, True)

    def encode(self, r: int, c: int, cols: int) -> int:
        return r * cols + c

    def setMemoAndReturn(self, memo: list[list[list[bool]]], catPos: int, mousePos: int, turn: int, value: bool) -> bool:
        memo[catPos][mousePos][turn] = value
        return value

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