<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>

def divide(n):
    t,f = 0,0
    while n % 5 == 0:
        n //= 5
        f += 1
    while n % 2 == 0:
        n //= 2
        t += 1
    return t,f


class Solution:
    def maxTrailingZeros(self, grid: List[List[int]]) -> int:
        n,m = len(grid),len(grid[0])
        pre2 = [ [0] * (m+1) for i in range(n+1) ]
        pre5 = [ [0] * (m+1) for i in range(n+1) ]
        for i in range(n):
            for j in range(m):
                t,f = divide(grid[i][j])
                pre2[i+1][j+1] = pre2[i+1][j] + pre2[i][j+1] - pre2[i][j] + t
                pre5[i+1][j+1] = pre5[i+1][j] + pre5[i][j+1] - pre5[i][j] + f
        
        res = 0
        for i in range(n):
            for j in range(m):
                top2,top5 = pre2[i+1][j+1] - pre2[i+1][j], pre5[i+1][j+1] - pre5[i+1][j]
                down2 = pre2[n][j+1] - pre2[n][j] - pre2[i][j+1] + pre2[i][j]
                down5 = pre5[n][j+1] - pre5[n][j] - pre5[i][j+1] + pre5[i][j]

                t,f = divide(grid[i][j])
                left2,left5 = pre2[i+1][j+1] - pre2[i][j+1] - t, pre5[i+1][j+1] - pre5[i][j+1] - f
                right2 = pre2[i+1][m] - pre2[i+1][j] - pre2[i][m] + pre2[i][j] - t 
                right5 = pre5[i+1][m] - pre5[i+1][j] - pre5[i][m] + pre5[i][j] - f 

                res = max(
                    res,
                    min(top2+left2,top5+left5),
                    min(top2+right2,top5+right5),
                    min(down2+left2,down5+left5),
                    min(down2+right2,down5+right5),
                )
        return res 

2772