<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>

int MOD = 1000000007;
class Solution {
    private:
        int iPs_[100000];
        int toIndex(int row, int col, int cols){
            return row * cols + col;
        }
        int incPaths(int begin_id, const vector<vector<int>>& grid){
            if(iPs_[begin_id] != -1){
                return iPs_[begin_id];
            }
            int row = begin_id / grid[0].size();
            int col = begin_id % grid[0].size();
            int val = grid[row][col];
            int ans = 1;
            if(row != 0){
                if(grid[row-1][col] > val){
                    ans = (ans + incPaths(toIndex(row-1, col, grid[0].size()), grid))%MOD;
                }
            }
            if(row != grid.size() - 1){
                if(grid[row+1][col] > val){
                    ans = (ans+incPaths(toIndex(row +1, col, grid[0].size()), grid))%MOD;
                }
            }
            if(col != 0){
                if(grid[row][col-1] > val){
                    ans = (ans+incPaths(toIndex(row, col-1, grid[0].size()), grid))%MOD;
                }
            }
            if(col != grid[0].size() - 1){
                if(grid[row][col + 1] > val){
                    ans = (ans+incPaths(toIndex(row, col + 1, grid[0].size()), grid))%MOD;
                }
            }
            return iPs_[begin_id] = ans;
        }
    public:
        int countPaths(vector<vector<int>>& grid) {
            int ans = 0;
            memset(iPs_, -1, sizeof(iPs_));
            for(int r = 0; r < grid.size(); ++r){
                for(int c= 0; c < grid[0].size(); ++c){
                    ans = (ans + incPaths(toIndex(r, c, grid[0].size()), grid))%MOD;
                }
            }
            return ans;
        }
};
43148