<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:
        rows , columns = len(grid) , len(grid[0])
        
        def count_factors(number, factor) :
            count = 0 
            while number % factor == 0 :
                number = number // factor
                count += 1
            
            return count
        
        two_factor = [[count_factors(grid[row][column], 2) for column in range(columns) ]for row in range(rows) ]
        five_factor =  [[count_factors(grid[row][column], 5) for column in range(columns) ]for row in range(rows) ]
        
        two_sum_h = [[0] * (columns + 1) for row in range(rows)]
        five_sum_h = [[0] * (columns + 1) for row in range(rows)]
        two_sum_v = [[0] * columns for row in range(rows+1)]
        five_sum_v = [[0] * columns for row in range(rows+1)]
        
        for row in range(rows) :
            for column in range(columns) :
                two_sum_h[row][column+1] = two_sum_h[row][column] + two_factor[row][column]
                five_sum_h[row][column+1] = five_sum_h[row][column] + five_factor[row][column]
                two_sum_v[row+1][column] = two_sum_v[row][column] + two_factor[row][column]
                five_sum_v[row+1][column] = five_sum_v[row][column] + five_factor[row][column]
                
        max_zeros = 0
        
        for row in range(rows) :
            for column in range(columns) :
                two_up = two_sum_v[row+1][column]
                five_up = five_sum_v[row+1][column]
                two_down = two_sum_v[rows][column] - two_sum_v[row][column]
                five_down = five_sum_v[rows][column] - five_sum_v[row][column]
                two_left = two_sum_h[row][column+1]
                five_left = five_sum_h[row][column+1]
                two_right =  two_sum_h[row][columns] - two_sum_h[row][column]
                five_right = five_sum_h[row][columns] - five_sum_h[row][column]
                
                # left ---> Up
                max_zeros = max(max_zeros , min(two_left+two_up-two_factor[row][column] , five_left+five_up-five_factor[row][column] ))
                # left ---> Down
                max_zeros = max(max_zeros , min(two_left+two_down-two_factor[row][column] , five_left+five_down-five_factor[row][column] ))
                 # right ---> Up
                max_zeros = max(max_zeros , min(two_right+two_up-two_factor[row][column] , five_right+five_up-five_factor[row][column] ))
                # right ---> Down
                max_zeros = max(max_zeros , min(two_right+two_down-two_factor[row][column] , five_right+five_down-five_factor[row][column] ))
                
        return max_zeros
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