<p>There is an <code>m x n</code> grid with a ball. The ball is initially at the position <code>[startRow, startColumn]</code>. You are allowed to move the ball to one of the four adjacent cells in the grid (possibly out of the grid crossing the grid boundary). You can apply <strong>at most</strong> <code>maxMove</code> moves to the ball.</p>

<p>Given the five integers <code>m</code>, <code>n</code>, <code>maxMove</code>, <code>startRow</code>, <code>startColumn</code>, return the number of paths to move the ball out of the grid boundary. Since the answer can be very large, return it <strong>modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/04/28/out_of_boundary_paths_1.png" style="width: 500px; height: 296px;" />
<pre>
<strong>Input:</strong> m = 2, n = 2, maxMove = 2, startRow = 0, startColumn = 0
<strong>Output:</strong> 6
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/04/28/out_of_boundary_paths_2.png" style="width: 500px; height: 293px;" />
<pre>
<strong>Input:</strong> m = 1, n = 3, maxMove = 3, startRow = 0, startColumn = 1
<strong>Output:</strong> 12
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= m, n &lt;= 50</code></li>
	<li><code>0 &lt;= maxMove &lt;= 50</code></li>
	<li><code>0 &lt;= startRow &lt; m</code></li>
	<li><code>0 &lt;= startColumn &lt; n</code></li>
</ul>

class Solution {
public:
    int M;
    int N;
    int MOD = 1e9+7;
    vector<vector<int>> directions{{0, 1}, {0, -1}, {1, 0}, {-1, 0}};
    unordered_map<string, int>t;
    int solve(int maxMove, int startRow, int startColumn) {
        if(startRow < 0 || startRow >= M || startColumn < 0 || startColumn >= N) {
            return 1; 
        }

        if(maxMove <= 0) 
            return 0;
        string temp = to_string(startRow) + "_"+ to_string(startColumn) +"_" + to_string(maxMove);

        if(t.find(temp)!=t.end()) {
            return t[temp];
        }

        int result = 0;
        for(auto &dir : directions) {
            int x = startRow    + dir[0];
            int y = startColumn + dir[1];
            
            result = (result + solve(maxMove-1, x, y)) % MOD;
        }
        

        return t[temp] =  result%MOD;

    }

    int findPaths(int m, int n, int maxMove, int startRow, int startColumn) {
        
        M = m;
        N = n;
        return solve(maxMove, startRow, startColumn);
    }
};

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