<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 {
    int dx[4] = {1, -1, 0, 0};
    int dy[4] = {0, 0, 1, -1};
    int dp[2][81][8][8][8][8];
    inline bool canGo(vector<string>& grid, int i , int j) {
        return i >= 0 && i < grid.size() && j >= 0 && j < grid[0].size()
        && grid[i][j] != '#';
    }

public:
    int solve(vector<string>& grid, int catJump, int mouseJump, int turn,
     int mouseSteps, int mousePosX, int mousePosY, int catPosX, int catPosY) {
        if (mouseSteps > 80) {
            return 0;
        }
        if (grid[mousePosX][mousePosY] == 'F') {
            return 1;
        }
        if (mousePosX == catPosX && mousePosY == catPosY) {
            return 0;
        }
        if (grid[catPosX][catPosY] == 'F') {
            return 0;
        }

        int &ans = dp[turn][mouseSteps][mousePosX][mousePosY][catPosX][catPosY];
        if (ans != -1) {
            return ans;
        }

        bool mouseCanWin = 0;
        if (turn) {
            mouseCanWin = 1;
            for (int k = 0; k < 4; k++) {
                for (int jump = 0; jump <= catJump; jump++) {
                    int newCatPosX = catPosX + dx[k] * jump;
                    int newCatPosY = catPosY + dy[k] * jump;
                    if (canGo(grid, newCatPosX, newCatPosY)) {
                        mouseCanWin &= solve(grid, catJump, mouseJump, 0, mouseSteps, mousePosX, mousePosY, newCatPosX, newCatPosY);
                    if (!mouseCanWin) break;
                    } else break; // cannot jump over the wall
                }
            }
        } else {
            mouseCanWin = 0;
            for (int k = 0; k < 4; k++) {
                for (int jump = 0; jump <= mouseJump; jump++) {
                    int newMousePosX = mousePosX + dx[k] * jump;
                    int newMousePosY = mousePosY + dy[k] * jump;
                    if (canGo(grid, newMousePosX, newMousePosY)) {
                        mouseCanWin |= solve(grid, catJump, mouseJump, 1, mouseSteps + 1, newMousePosX, newMousePosY, catPosX, catPosY);
                        if (mouseCanWin) break;
                    } else break; // cannot jump over the wall               
                }
            }
        }
        return ans = mouseCanWin;
    }

    bool canMouseWin(vector<string>& grid, int catJump, int mouseJump) {
        memset(dp, -1, sizeof(dp));
        int mousePosX = 0;
        int mousePosY = 0;
        int catPosX = 0;
        int catPosY = 0;
        for (int i = 0; i < grid.size(); i++) {
            for (int j = 0; j < grid[0].size(); j++) {
                if (grid[i][j] == 'C') {
                    catPosX = i;
                    catPosY = j;
                }
                if (grid[i][j] == 'M') {
                    mousePosX = i;
                    mousePosY = j;
                }
            }
        }
        return solve(grid, catJump, mouseJump, 0, 0, mousePosX, mousePosY, catPosX, catPosY);
    }
};
14500