<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 findPaths(int m, int n, int maxMove, int startRow, int startColumn) {
    vector<vector<vector<int>>> mem(maxMove + 1,
                                    vector<vector<int>>(m, vector<int>(n, -1)));
    return findPaths(m, n, maxMove, startRow, startColumn, mem);
  }

 private:
  static constexpr int kMod = 1'000'000'007;

  // Returns the number of paths to move the ball at (i, j) out-of-bounds with k
  // moves.
  int findPaths(int m, int n, int k, int i, int j,
                vector<vector<vector<int>>>& mem) {
    if (i < 0 || i == m || j < 0 || j == n)
      return 1;
    if (k == 0)
      return 0;
    if (mem[k][i][j] != -1)
      return mem[k][i][j];
    return mem[k][i][j] = (findPaths(m, n, k - 1, i + 1, j, mem) * 1L +
                           findPaths(m, n, k - 1, i - 1, j, mem) +
                           findPaths(m, n, k - 1, i, j + 1, mem) +
                           findPaths(m, n, k - 1, i, j - 1, mem)) %
                          kMod;
  }
};
11286