<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_2or5(num):
            cnt2 = 0
            cnt5 = 0
            while bin(num)[-1] == '0':
                num >>= 1
                cnt2 += 1
            while num % 5 == 0:
                num = num // 5
                cnt5 += 1
            return cnt2, cnt5
        m = len(grid)
        n = len(grid[0])
        store = [[(0,0) for _ in range(n)] for _ in range(m)]
        for i in range(m):
            for j in range(n):
                item = grid[i][j]
                store[i][j] = count_2or5(item)

        row_cnt = [[(0,0) for _ in range(n)] for _ in range(m)]
        col_cnt = [[(0,0) for _ in range(n)] for _ in range(m)]
        for i in range(m):
            for j in range(n):
                if j == 0:
                    row_cnt[i][j] = store[i][j]
                    continue
                row_cnt[i][j] = (row_cnt[i][j-1][0] + store[i][j][0], row_cnt[i][j-1][1] + store[i][j][1])
        for i in range(m):
            for j in range(n):
                if i == 0:
                    col_cnt[i][j] = store[i][j]
                    continue
                col_cnt[i][j] = (col_cnt[i-1][j][0] + store[i][j][0], col_cnt[i-1][j][1] + store[i][j][1])

        res = 0
        for i in range(m):
            for j in range(n):
                left = row_cnt[i][j]
                right = (row_cnt[i][n-1][0] - row_cnt[i][j][0], row_cnt[i][n-1][1] - row_cnt[i][j][1])
                up = col_cnt[i][j]
                down = (col_cnt[m-1][j][0] - col_cnt[i][j][0], col_cnt[m-1][j][1] - col_cnt[i][j][1])

                res = max(res, min(left[0]+up[0] - store[i][j][0], left[1]+up[1] - store[i][j][1]), 
                                min(left[0]+down[0], left[1]+down[1]),
                                min(right[0]+up[0], right[1]+up[1]), 
                                min(right[0]+down[0] + store[i][j][0], right[1]+down[1] + store[i][j][1]),
                                )
        return res

    
2577