<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>

class Solution:
    mod = int(1e9) + 7

    def checkLimits(self, i, j, n, m):
        return i >= 0 and i < n and j >= 0 and j < m

    def func(self, matrix, dp, i, j, n, m):
        if not self.checkLimits(i, j, n, m):
            return 0

        if dp[i][j] != -1:
            return dp[i][j]

        c1 = c2 = c3 = c4 = 0

        if self.checkLimits(i, j + 1, n, m) and matrix[i][j + 1] > matrix[i][j]:
            c1 = self.func(matrix, dp, i, j + 1, n, m)

        if self.checkLimits(i, j - 1, n, m) and matrix[i][j - 1] > matrix[i][j]:
            c2 = self.func(matrix, dp, i, j - 1, n, m)

        if self.checkLimits(i + 1, j, n, m) and matrix[i + 1][j] > matrix[i][j]:
            c3 = self.func(matrix, dp, i + 1, j, n, m)

        if self.checkLimits(i - 1, j, n, m) and matrix[i - 1][j] > matrix[i][j]:
            c4 = self.func(matrix, dp, i - 1, j, n, m)

        dp[i][j] = (1 + c1 + c2 + c3 + c4) % self.mod

        return dp[i][j]

    def countPaths(self, matrix):
        row = len(matrix)
        col = len(matrix[0])
        dp = [[-1] * col for _ in range(row)]
        ans = 0

        for i in range(row):
            for j in range(col):
                ans += self.func(matrix, dp, i, j, row, col)
                ans %= self.mod

        return ans
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