<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:
    dirs = ((0, 1), (1, 0), (0, -1), (-1, 0))
    m = len(grid)
    n = len(grid[0])
    nFloors = 0
    cat = 0  # cat's position
    mouse = 0  # mouse's position

    def hash(i: int, j: int) -> int:
      return i * n + j

    for i in range(m):
      for j in range(n):
        if grid[i][j] != '#':
          nFloors += 1
        if grid[i][j] == 'C':
          cat = hash(i, j)
        elif grid[i][j] == 'M':
          mouse = hash(i, j)

    @functools.lru_cache(None)
    def dp(cat: int, mouse: int, turn: int) -> bool:
      """
      Returns True if the mouse can win, where the cat is on (i / 8, i % 8), the
      mouse is on (j / 8, j % 8), and the turns is k.
      """
      # We already search the whole touchable grid.
      if turn == nFloors * 2:
        return False

      if turn % 2 == 0:
        # the mouse's turn
        i = mouse // n
        j = mouse % n
        for dx, dy in dirs:
          for jump in range(mouseJump + 1):
            x = i + dx * jump
            y = j + dy * jump
            if x < 0 or x == m or y < 0 or y == n:
              break
            if grid[x][y] == '#':
              break
            # The mouse eats the food, so the mouse wins.
            if grid[x][y] == 'F':
              return True
            if dp(cat, hash(x, y), turn + 1):
              return True
        # The mouse can't win, so the mouse loses.
        return False
      else:
        # the cat's turn
        i = cat // n
        j = cat % n
        for dx, dy in dirs:
          for jump in range(catJump + 1):
            x = i + dx * jump
            y = j + dy * jump
            if x < 0 or x == m or y < 0 or y == n:
              break
            if grid[x][y] == '#':
              break
            # The cat eats the food, so the mouse loses.
            if grid[x][y] == 'F':
              return False
            nextCat = hash(x, y)
            # The cat catches the mouse, so the mouse loses.
            if nextCat == mouse:
              return False
            if not dp(nextCat, mouse, turn + 1):
              return False
        # The cat can't win, so the mouse wins.
        return True

    return dp(cat, mouse, 0)
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