<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:
    vector<vector<int>> directions = {{0, 1}, {0, -1}, {1, 0}, {-1, 0}};
    int m;
    int n;
    long long modulo = 1000000007;

    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)));
        this->m = m;
        this->n = n;
        return helper(dp, startRow, startColumn, maxMove);
    }

    int helper(vector<vector<vector<int>>>& dp,
               int row,
               int col,
               int remainingMoves)
    {
        if (row < 0 || row >= m || col < 0 || col >= n)
        {
            return 1;
        }
        else if (remainingMoves == 0)
        {
            return 0;
        }
        else if (dp[row][col][remainingMoves] != -1)
        {
            return dp[row][col][remainingMoves];
        }
        else
        {
            int result = 0;
            for (const vector<int> direction : directions)
            {
                int new_row = row + direction[0];
                int new_col = col + direction[1];
                result = (result + helper(dp, new_row, new_col, remainingMoves - 1)) % modulo;
            }
            dp[row][col][remainingMoves] = result;
            return result;
        }
    }
};
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