<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>


constexpr int MOD = 1e9 + 7;

using Point = std::pair<int, int>;

using Key = std::pair<Point, int>;

namespace std {
    template <>
    struct hash<Key> {
        size_t operator()(const Key& k) const {
            return hash<int>()(k.first.first) ^ hash<int>()(k.first.second) ^ hash<int>()(k.second);
        }
    };
}

vector<Point> neighbors(const Point p){
    return {{p.first-1, p.second}, {p.first+1, p.second}, {p.first, p.second-1}, {p.first, p.second+1}};
}

bool is_out_of_border(const Point p, const int x_num, const int y_num){
    return p.first < 0 || p.first >= x_num || p.second < 0 || p.second >= y_num;
}

unordered_map<Key, int> cache;

bool has_cache(const Point p, const int maxMove){
    return cache.find({p, maxMove}) != cache.end();
}

int get_cache(const Point p, const int maxMove){
    return cache[{p, maxMove}];
}

void set_cache(const Point p, const int maxMove, const int value){
    cache[{p, maxMove}] = value;
}

int mod_add(const int a, const int b){
    return (a + b) % MOD;
}

int dfs(const Point p, const int x_num, const int y_num, const int maxMove){
    assert(maxMove >= 0);
    if(is_out_of_border(p, x_num, y_num)){
        return 1;
    }
    if (maxMove == 0){
        return 0;
    }
    if (has_cache(p, maxMove)){
        return get_cache(p, maxMove);
    }
    int res = 0;
    for(const auto& neighbor : neighbors(p)){
        res = mod_add(res, dfs(neighbor, x_num, y_num, maxMove-1));
    }
    set_cache(p, maxMove, res);
    cout << "p: " << p.first << ", " << p.second << ", " << maxMove << ", res: " << res << endl;
    return res;
}

class Solution {
public:
    int findPaths(int m, int n, int maxMove, int startRow, int startColumn) {
        cache.clear();
        const int res = dfs({startRow, startColumn}, m, n, maxMove);
        return res;
    }
};

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