<p>You are given a <code>m x n</code> matrix <code>grid</code>. Initially, you are located at the top-left corner <code>(0, 0)</code>, and in each step, you can only <strong>move right or down</strong> in the matrix.</p>

<p>Among all possible paths starting from the top-left corner <code>(0, 0)</code> and ending in the bottom-right corner <code>(m - 1, n - 1)</code>, find the path with the <strong>maximum non-negative product</strong>. The product of a path is the product of all integers in the grid cells visited along the path.</p>

<p>Return the <em>maximum non-negative product <strong>modulo</strong> </em><code>10<sup>9</sup> + 7</code>. <em>If the maximum product is <strong>negative</strong>, return </em><code>-1</code>.</p>

<p>Notice that the modulo is performed after getting the maximum product.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/12/23/product1.jpg" style="width: 244px; height: 245px;" />
<pre>
<strong>Input:</strong> grid = [[-1,-2,-3],[-2,-3,-3],[-3,-3,-2]]
<strong>Output:</strong> -1
<strong>Explanation:</strong> It is not possible to get non-negative product in the path from (0, 0) to (2, 2), so return -1.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/12/23/product2.jpg" style="width: 244px; height: 245px;" />
<pre>
<strong>Input:</strong> grid = [[1,-2,1],[1,-2,1],[3,-4,1]]
<strong>Output:</strong> 8
<strong>Explanation:</strong> Maximum non-negative product is shown (1 * 1 * -2 * -4 * 1 = 8).
</pre>

<p><strong class="example">Example 3:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/12/23/product3.jpg" style="width: 164px; height: 165px;" />
<pre>
<strong>Input:</strong> grid = [[1,3],[0,-4]]
<strong>Output:</strong> 0
<strong>Explanation:</strong> Maximum non-negative product is shown (1 * 0 * -4 = 0).
</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;= 15</code></li>
	<li><code>-4 &lt;= grid[i][j] &lt;= 4</code></li>
</ul>

#define mod 1000000007
#define ll long long 
class Solution {
private:
    pair<ll,ll> solve(vector<vector<int>>&grid,int i,int j,vector<vector<pair<ll,ll>>>&dp)
    {
        auto [x,y]=dp[i][j];
        if(x!=-1 && y!=-1)   return dp[i][j];
        if(i==0 && j==0)  return {grid[i][j],grid[i][j]};
        if(i==0)
        {
            auto [maxi,mini]=solve(grid,i,j-1,dp);
            return dp[i][j]={maxi*grid[i][j],maxi*grid[i][j]};
        }
        else if(j==0)
        {
            auto [maxi,mini]=solve(grid,i-1,j,dp);
            return dp[i][j]={maxi*grid[i][j],maxi*grid[i][j]};
        }
        auto [umaxi,umini]=solve(grid,i-1,j,dp);
        auto [lmaxi,lmini]=solve(grid,i,j-1,dp);
        ll maxi=max({grid[i][j]*umaxi,grid[i][j]*umini,grid[i][j]*lmaxi,grid[i][j]*lmini});
        ll mini=min({grid[i][j]*umaxi,grid[i][j]*umini,grid[i][j]*lmaxi,grid[i][j]*lmini});
        return dp[i][j]={maxi,mini};
    }
public:
    int maxProductPath(vector<vector<int>>& grid) {
        int m = grid.size();
        int n = grid[0].size();
        vector<vector<pair<ll,ll>>> dp(m, vector<pair<ll,ll>>(n, {-1, -1}));
        auto [maxi, mini] = solve(grid, m-1, n-1, dp);
        if(maxi<0)   return -1;
        return maxi%mod;
    }
};
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