<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:
    mod = int(1e9 + 7)

    def __init__(self):
        self.memo = [[[-1 for _ in range(51)] for _ in range(51)] for _ in range(51)]

    def dfs(self, m, n, maxMove, startRow, startColumn):
        if startRow < 0 or startColumn < 0 or startRow >= m or startColumn >= n:
            return 1
        if maxMove == 0:
            return 0
        if self.memo[startRow][startColumn][maxMove] != -1:
            return self.memo[startRow][startColumn][maxMove]
        
        up = self.dfs(m, n, maxMove - 1, startRow - 1, startColumn)
        down = self.dfs(m, n, maxMove - 1, startRow + 1, startColumn)
        left = self.dfs(m, n, maxMove - 1, startRow, startColumn - 1)
        right = self.dfs(m, n, maxMove - 1, startRow, startColumn + 1)
        
        self.memo[startRow][startColumn][maxMove] = ((up + down) % self.mod + (left + right) % self.mod) % self.mod
        return self.memo[startRow][startColumn][maxMove]

    def findPaths(self, m, n, maxMove, startRow, startColumn):
        self.memo = [[[-1 for _ in range(51)] for _ in range(51)] for _ in range(51)]
        return self.dfs(m, n, maxMove, startRow, startColumn)


19000