<p>You are given an <code>m x n</code> integer matrix <code>grid</code>, where you can move from a cell to any adjacent cell in all <code>4</code> directions.</p>

<p>Return <em>the number of <strong>strictly</strong> <strong>increasing</strong> paths in the grid such that you can start from <strong>any</strong> cell and end at <strong>any</strong> cell. </em>Since the answer may be very large, return it <strong>modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>

<p>Two paths are considered different if they do not have exactly the same sequence of visited cells.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/05/10/griddrawio-4.png" style="width: 181px; height: 121px;" />
<pre>
<strong>Input:</strong> grid = [[1,1],[3,4]]
<strong>Output:</strong> 8
<strong>Explanation:</strong> The strictly increasing paths are:
- Paths with length 1: [1], [1], [3], [4].
- Paths with length 2: [1 -&gt; 3], [1 -&gt; 4], [3 -&gt; 4].
- Paths with length 3: [1 -&gt; 3 -&gt; 4].
The total number of paths is 4 + 3 + 1 = 8.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> grid = [[1],[2]]
<strong>Output:</strong> 3
<strong>Explanation:</strong> The strictly increasing paths are:
- Paths with length 1: [1], [2].
- Paths with length 2: [1 -&gt; 2].
The total number of paths is 2 + 1 = 3.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>m == grid.length</code></li>
	<li><code>n == grid[i].length</code></li>
	<li><code>1 &lt;= m, n &lt;= 1000</code></li>
	<li><code>1 &lt;= m * n &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= grid[i][j] &lt;= 10<sup>5</sup></code></li>
</ul>

"""
Scratch:
Count number of paths starting at each cell
f(cell) = 1 + sum(f(greater neighbor of cell))
"""
MOD = 10 ** 9 + 7
class Solution:
    def countPaths(self, grid: List[List[int]]) -> int:
        self.m, self.n = len(grid), len(grid[0])
        self.grid = grid
        self.cache = [[0 for _ in range(self.n)] for _ in range(self.m)]
        self.total = 0
        for r in range(self.m):
            for c in range(self.n):
                self.visit(r, c)
        return self.total
        
    def visit(self, row, col) -> int:
        if self.cache[row][col]:
            return self.cache[row][col]

        curr = 1
        for nr, nc in self.greater_neighbors(row, col):
            curr += self.visit(nr, nc)
        self.cache[row][col] = curr
        self.total = (self.total + curr) % MOD
        return curr

    def greater_neighbors(self, row, col) -> List[Tuple[int, int]]:
        result = []
        dirs = [(-1, 0), (1, 0), (0, 1), (0, -1)]
        for d in dirs:
            nr, nc = row + d[0], col + d[1]
            if self.in_bounds(nr, nc) and self.grid[nr][nc] > self.grid[row][col]:
                result.append((nr, nc))
        return result

    def in_bounds(self, nr, nc):
        return 0 <= nr < self.m and 0 <= nc < self.n

        
        
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