<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>

const int mod = 1e9 + 7;  
class Solution {
    int solve(int m , int n , int maxMove , int row , int col , vector<vector<vector<int>>> &dp) {
        // Base Case:
        // 1. we have no more moves so no way to discover new paths.
        // 2. We have out of bounds , so 1 valid paths
        if(maxMove < 0) return 0; 
        if(col < 0 || row < 0 || col >= n || row >= m) return 1;
        
        // check memoization part
        if(dp[row][col][maxMove] != -1) return dp[row][col][maxMove];

        vector<vector<int>> direction = {{-1,0} , {1,0} , {0,-1} , {0,1}};

        int ans = 0;
        for(int i = 0; i < 4; i++) {
            int currRow = row + direction[i][0];
            int currCol = col + direction[i][1];

            ans = (ans + solve(m , n , maxMove - 1 , currRow , currCol , dp)) % mod;
        }

        dp[row][col][maxMove] = ans;
       return ans; 
     } 
public:
    int findPaths(int m, int n, int maxMove, int startRow, int startColumn) {
        vector<vector<vector<int>>> dp(m , (vector<vector<int>>(n , vector<int>(maxMove+1 , -1))));
        int ans = solve(m , n , maxMove , startRow , startColumn , dp);
       return ans; 
    }
};
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