<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:
    bool canMouseWin(vector<string>& g, int catJump, int mouseJump) {
        int c_pos , m_pos, f_pos, n = g.size()*g[0].size();
        for(int i = 0; i < n; i++) {
            switch (g[i/g[0].size()][i%g[0].size()]) {
                case 'C': c_pos = i; break;
                case 'F': f_pos = i; break;
                case 'M': m_pos = i; break;
            }
        }

        int limits = 64*2, dirs[5] = {0, 1, 0, -1, 0};
        vector<vector<vector<int>>> memo(n, vector<vector<int>>(n,vector<int>(limits+1, -1)));
        function<bool(int, int, int)> dfs = [&](int c_pos, int m_pos, int step) {
            bool isCat = step&1;
            if(c_pos == f_pos || m_pos == c_pos || step >= limits) return isCat;
            if(m_pos == f_pos) return !isCat;

            if(memo[c_pos][m_pos][step] != -1) return (bool)memo[c_pos][m_pos][step];
            int pos = isCat ? c_pos : m_pos, jump = isCat ? catJump : mouseJump;
            int x = pos/g[0].size(), y = pos%g[0].size();
            for(int d = 0; d < 4; d++) {
                for(int i = 0; i <= jump; i++) {//i = 0, stay put
                    int nxt_x = x + i*dirs[d], nxt_y = y + i*dirs[d+1];
                    if(nxt_x < 0 || nxt_y < 0 || nxt_x >= g.size() || nxt_y >= g[0].size() || g[nxt_x][nxt_y] == '#') break;
                    int nxt_pos = nxt_x*g[0].size()+nxt_y;
                    if(!dfs(isCat? nxt_pos : c_pos, isCat ? m_pos : nxt_pos, step+1)) {
                        memo[c_pos][m_pos][step] = true;
                        return true;
                    }
                }
            }
            memo[c_pos][m_pos][step] = false;
            return false;
        };
        return dfs(c_pos, m_pos, 0);
    }
};
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