<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<long long> delx = {1, -1, 0, 0};
    vector<long long> dely = {0, 0, 1, -1};
    const long long MOD = 1e9 + 7;

    long long f(long long x, long long y, long long n, long long m, long long maxmove, map<pair<pair<long long, long long>, long long>, long long> &dp) {
        if(dp.find({{x, y}, maxmove}) != dp.end()) 
            return dp[{{x, y}, maxmove}];
        
        long long res = 0;
        for(long long i = 0; i < 4; i++) {
            long long nx = x + delx[i];
            long long ny = y + dely[i];
            if(nx >= 0 && ny >= 0 && nx < n && ny < m && maxmove > 0) {
                res = (res + f(nx, ny, n, m, maxmove - 1, dp)) % MOD;
            } else if((nx < 0 || ny < 0 || nx >= n || ny >= m) && maxmove > 0) {
                res = (res + 1) % MOD;
            }
        }

        return dp[{{x, y}, maxmove}] = res;
    }

    long long findPaths(long long m, long long n, long long maxMove, long long startRow, long long startColumn) {
        map<pair<pair<long long, long long>, long long>, long long> dp;
        return f(startRow, startColumn, m, n, maxMove, dp);
    }
};

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