<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>

MOD = int(1e9 + 7)
class Solution:
    def countPaths(self, grid: List[List[int]]) -> int:
        # the state: (x, y)
        # the output: the number of path can go
        # the transition: adjacent
        dx = [1, -1, 0, 0]
        dy = [0, 0, 1, -1]

        m = len(grid)
        n = len(grid[0])
        def checkIsValid(x, y):
            return 0 <= x < m and 0 <= y < n
        memorize = dict()
        def countPath(x, y, visited):
            if (x, y) in memorize:
                return memorize[(x, y)]
            
            cnt = 1
            for i in range(4):
                ux = x + dx[i]
                uy = y + dy[i]
                if checkIsValid(ux, uy) and (ux, uy) not in visited and grid[ux][uy] > grid[x][y]:
                    visited.add((ux, uy))
                    cnt += int(countPath(ux, uy, visited) % MOD)
                    visited.remove((ux, uy))
            memorize[(x, y)] = int(cnt % MOD)
            return cnt
        
        ans = 0
        for i in range(m):
            for j in range(n):
                visited = set()
                visited.add((i, j))
                ans += int(countPath(i, j, visited) % MOD)
        return int(ans % MOD)

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