<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>

#include <vector>
#include <cstring>
#include <string>

using namespace std;

class Solution {
public:
    int dp[10][10][10][10][1010];
    vector<int> dx = {-1, 0, 1, 0};  
    vector<int> dy = {0, 1, 0, -1};

    int solve(int turn, pair<int, int> mouse, pair<int, int> cat, vector<string>& grid, int catJump, int mouseJump) {
        if (turn >= 1000) return 2;  
        if (dp[mouse.first][mouse.second][cat.first][cat.second][turn] != -1)
            return dp[mouse.first][mouse.second][cat.first][cat.second][turn];

        int mouseX = mouse.first, mouseY = mouse.second;
        int catX = cat.first, catY = cat.second;

        if (grid[mouseX][mouseY] == 'F') return dp[mouseX][mouseY][catX][catY][turn] = 1;  
        if (grid[catX][catY] == 'F' || (mouseX == catX && mouseY == catY))
            return dp[mouseX][mouseY][catX][catY][turn] = 2;  

        if (turn % 2 == 0) {  
            for (int i = 0; i < 4; ++i) {
                for (int j = 0; j <= mouseJump; ++j) {
                    int newX = mouseX + dx[i] * j;
                    int newY = mouseY + dy[i] * j;

                    if (newX < 0 || newY < 0 || newX >= grid.size() || newY >= grid[0].size())
                        continue;
                    if(grid[newX][newY] == '#')break;

                    if (solve(turn + 1, {newX, newY}, cat, grid, catJump, mouseJump) == 1)
                        return dp[mouseX][mouseY][catX][catY][turn] = 1;
                }
            }
            return dp[mouseX][mouseY][catX][catY][turn] = 2;
        } else { 
            for (int i = 0; i < 4; ++i) {
                for (int j = 0; j <= catJump; ++j) {
                    int newX = catX + dx[i] * j;
                    int newY = catY + dy[i] * j;

                   if (newX < 0 || newY < 0 || newX >= grid.size() || newY >= grid[0].size())
                        continue;
                    if(grid[newX][newY] == '#')break;
                    if (solve(turn + 1, mouse, {newX, newY}, grid, catJump, mouseJump) == 2)
                        return dp[mouseX][mouseY][catX][catY][turn] = 2;
                }
            }
            return dp[mouseX][mouseY][catX][catY][turn] = 1;
        }
    }

    bool canMouseWin(vector<string>& grid, int catJump, int mouseJump) {
        memset(dp, -1, sizeof(dp));
        pair<int, int> mouse, cat;

        for (int i = 0; i < grid.size(); ++i) {
            for (int j = 0; j < grid[0].size(); ++j) {
                if (grid[i][j] == 'M') mouse = {i, j};
                if (grid[i][j] == 'C') cat = {i, j};
            }
        }

        return solve(0, mouse, cat, grid, catJump, mouseJump) == 1;
    }
};

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