<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:
        m, n = len(grid), len(grid[0])
        init_pos = {
            grid[r][c]: (r, c)
            for r in range(m) for c in range(n)
            if grid[r][c] in 'MCF'
        }
        
        def next_positions(r, c, jump):
            yield r, c  # can stay still
            for dr, dc in [(0, 1), (0, -1), (1, 0), (-1, 0)]:
                for k in range(1, jump + 1):
                    rr, cc = r + k * dr, c + k * dc
                    if not (0 <= rr < m and 0 <= cc < n):
                        break
                    if grid[rr][cc] == '#':
                        break
                    yield rr, cc
        
        cache, G_mouse, G_cat = defaultdict(int), {}, {}
        queue = deque()
        
        for r in range(m):
            for c in range(n):
                if grid[r][c] == '#':
                    continue
                for turn in 'MC':
                    cache[((r, c), (r, c), turn)] = 2
                    cache[((r, c), init_pos['F'], turn)] = 2
                    cache[(init_pos['F'], (r, c), turn)] = 1
                    queue.append((init_pos['F'], (r, c), turn))
                
                G_mouse[(r, c)] = set(next_positions(r, c, mouseJump))
                G_cat[(r, c)] = set(next_positions(r, c, catJump))
        
        while queue:
            pos_mouse, pos_cat, turn = queue.popleft()
            if turn == 'M':  # Last time move by mouse
                for last_pos_mouse in G_mouse[pos_mouse]:
                    last_state = (last_pos_mouse, pos_cat, 'C')
                    if cache[last_state] > 0:
                        continue
                    cache[last_state] = 1
                    queue.append(last_state)
            else:  # Last time move by cat
                for last_pos_cat in G_cat[pos_cat]:
                    last_state = (pos_mouse, last_pos_cat, 'M')
                    if cache[last_state] > 0:
                        continue
                    cache[last_state] -= 1
                    if -cache[last_state] == len(G_cat[last_pos_cat]):
                        cache[last_state] = 1
                        queue.append(last_state)
        
        init_state = (init_pos['M'], init_pos['C'], 'C')
        return cache[init_state] == 1
17100