<p>You are given a 2D integer array <code>grid</code> of size <code>m x n</code>, where each cell contains a positive integer.</p>

<p>A <strong>cornered path</strong> is defined as a set of adjacent cells with <strong>at most</strong> one turn. More specifically, the path should exclusively move either <strong>horizontally</strong> or <strong>vertically</strong> up to the turn (if there is one), without returning to a previously visited cell. After the turn, the path will then move exclusively in the <strong>alternate</strong> direction: move vertically if it moved horizontally, and vice versa, also without returning to a previously visited cell.</p>

<p>The <strong>product</strong> of a path is defined as the product of all the values in the path.</p>

<p>Return <em>the <strong>maximum</strong> number of <strong>trailing zeros</strong> in the product of a cornered path found in </em><code>grid</code>.</p>

<p>Note:</p>

<ul>
	<li><strong>Horizontal</strong> movement means moving in either the left or right direction.</li>
	<li><strong>Vertical</strong> movement means moving in either the up or down direction.</li>
</ul>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/03/23/ex1new2.jpg" style="width: 577px; height: 190px;" />
<pre>
<strong>Input:</strong> grid = [[23,17,15,3,20],[8,1,20,27,11],[9,4,6,2,21],[40,9,1,10,6],[22,7,4,5,3]]
<strong>Output:</strong> 3
<strong>Explanation:</strong> The grid on the left shows a valid cornered path.
It has a product of 15 * 20 * 6 * 1 * 10 = 18000 which has 3 trailing zeros.
It can be shown that this is the maximum trailing zeros in the product of a cornered path.

The grid in the middle is not a cornered path as it has more than one turn.
The grid on the right is not a cornered path as it requires a return to a previously visited cell.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/03/25/ex2.jpg" style="width: 150px; height: 157px;" />
<pre>
<strong>Input:</strong> grid = [[4,3,2],[7,6,1],[8,8,8]]
<strong>Output:</strong> 0
<strong>Explanation:</strong> The grid is shown in the figure above.
There are no cornered paths in the grid that result in a product with a trailing zero.
</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;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= m * n &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= grid[i][j] &lt;= 1000</code></li>
</ul>

class Solution:
    def maxTrailingZeros(self, grid: list[list[int]]) -> int:
        m, n = len(grid)+1, len(grid[0])+1
        grid = [[(0,0,0,0)]*n]+[[(0,0,0,0)]+row for row in grid]

        def prefix_sum(row,col):
            val = grid[row][col]
            factor_2, factor_5 = 0, 0

            while val > 0:
                if val%2: 
                    break
                val//= 2
                factor_2 += 1
   
            while val > 0:
                if val%5: 
                    break
                val//= 5
                factor_5 += 1

            u2, u5, _, _ = grid[row - 1][col]
            _, _, l2, l5 = grid[row][col - 1]

            return (factor_2 + u2, factor_5 + u5, factor_2 + l2, factor_5 + l5)
        
        def countZeros(r: int,c: int)-> int:
            up2   ,up5    = grid[r][c][0],grid[r][c][1]
            down2 ,down5  = grid[m-1][c][0]-grid[r-1][c][0],grid[m-1][c][1]-grid[r-1][c][1]
            
            left2 ,left5  = grid[r][c-1][2],grid[r][c-1][3]
            right2,right5 = grid[r][n-1][2]-grid[r][c][2],grid[r][n-1][3]-grid[r][c][3] 

            return max(min(up2+left2 ,up5+left5 ), min(down2+left2 ,down5+left5 ),
                       min(up2+right2,up5+right5), min(down2+right2,down5+right5))

        for r in range(1,m):
            for c in range(1,n):grid[r][c] = prefix_sum(r,c)
        
        return max(countZeros(r,c) for c in range(1,n) for r in range(1,m))
2217