<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 nRows, nCols;
    unordered_map<string, int> memo; 
    unordered_set<string> complete;
    int mod = 1000000007;
    vector<pair<int, int>> diff {
        {0, -1}, {0, 1}, {-1, 0}, {1, 0}
    };

    int findPaths(int m, int n, int maxMove, int startRow, int startColumn) {
        nRows = m, nCols = n;
        return countPaths(startRow, startColumn, maxMove);
    }

    int countPaths(int row, int col, int movesRem) {
        if (row < 0 || row >= nRows || col < 0 || col >= nCols) return 1;
        if (movesRem == 0) return 0;

        string key = to_string(row) + " " + to_string(col) + " " + to_string(movesRem);
        if (memo.find(key) != memo.end()) return memo[key];

        int ans = 0;
        for (auto [rDiff, cDiff] : diff) {
            ans = add(ans, countPaths(row + rDiff, col + cDiff, movesRem - 1));
        }
        return memo[key] = ans;
    }

    int add(int x, int y) {
        long long z = 1LL * x + 1LL * y;
        return z % mod;
    }

};
18858