<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:

        def count_div(target, div):
            count = 0
            while target > 0 and target % div == 0:
                target = target // div
                count += 1
            return count
        count_2 = [[count_div(target, 2) for target in row] for row in grid]
        count_5 = [[count_div(target, 5) for target in row] for row in grid]
        count_2_left = [[0] * len(grid[0]) for row in grid]
        count_2_right = [[0] * len(grid[0]) for row in grid]
        count_2_up = [[0] * len(grid[0]) for row in grid]
        count_2_down = [[0] * len(grid[0]) for row in grid]
        count_5_left = [[0] * len(grid[0]) for row in grid]
        count_5_right = [[0] * len(grid[0]) for row in grid]
        count_5_up = [[0] * len(grid[0]) for row in grid]
        count_5_down = [[0] * len(grid[0]) for row in grid]


        for r in range(len(grid)):
            left_2 = left_5 = 0
            for c in range(len(grid[0])):
                left_2 += count_2[r][c]
                left_5 += count_5[r][c]
                count_2_left[r][c] = left_2
                count_5_left[r][c] = left_5
        for r in range(len(grid)):
            right_2 = right_5 = 0
            for c in range(len(grid[0]) - 1, -1, -1):
                right_2 += count_2[r][c]
                right_5 += count_5[r][c]
                count_2_right[r][c] = right_2
                count_5_right[r][c] = right_5
        for c in range(len(grid[0])):
            up_2 = up_5 = 0
            for r in range(len(grid)):
                up_2 += count_2[r][c]
                up_5 += count_5[r][c]
                count_2_up[r][c] = up_2
                count_5_up[r][c] = up_5
        for c in range(len(grid[0])):
            down_2 = down_5 = 0
            for r in range(len(grid) - 1, -1, -1):
                down_2 += count_2[r][c]
                down_5 += count_5[r][c]
                count_2_down[r][c] = down_2
                count_5_down[r][c] = down_5
        result = 0
        for r in range(len(grid)):
            for c in range(len(grid[0])):
                left_up = min(count_5_left[r][c] + count_5_up[r][c] - count_5[r][c], count_2_left[r][c] + count_2_up[r][c] - count_2[r][c])
                left_down = min(count_5_left[r][c] + count_5_down[r][c] - count_5[r][c], count_2_left[r][c] + count_2_down[r][c] - count_2[r][c])
                right_up = min(count_5_right[r][c] + count_5_up[r][c] - count_5[r][c], count_2_right[r][c] + count_2_up[r][c] - count_2[r][c])
                right_down = min(count_5_right[r][c] + count_5_down[r][c] - count_5[r][c], count_2_right[r][c] + count_2_down[r][c] - count_2[r][c])
                result = max(result, left_up, left_down, right_up, right_down)
        return result
        



        
2708