<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>

import numpy as np


class Solution:
    def countPaths(self, grid: List[List[int]]) -> int:
        n_row = len(grid)
        n_col = len(grid[0])

        dp = np.ones((n_row, n_col), dtype=int) * -1

        offset_list = ((0, 1), (0, -1), (1, 0), (-1, 0))

        mod = 7 + 10**9

        def _dfs(i, j, dp):
            # calculate #paths that end at (i, j)
            if dp[i, j] != -1:
                return dp[i, j]
            
            res = 1
            for offset_i, offset_j in offset_list:
                new_i = i + offset_i
                new_j = j + offset_j

                if (new_i >= 0) and (new_i < n_row) and (new_j >= 0) and (new_j < n_col) and (grid[new_i][new_j] < grid[i][j]):
                    res += _dfs(new_i, new_j, dp) % mod
            
            dp[i, j] = res
            return res
        
        res = 0
        for i in range(n_row):
            for j in range(n_col):
                res += _dfs(i, j, dp) % mod
        
        return res % mod


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