<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, M: List[List[int]]) -> int:
        """prefix 2, 5"""
        m = len(M)
        n = len(M[0])

        for i in range(m):
            for j in range(n):
                c2 = 0
                q, r = divmod(M[i][j], 2)
                while r == 0:
                    c2 += 1
                    M[i][j] = q
                    q, r = divmod(M[i][j], 2)

                c5 = 0
                q, r = divmod(M[i][j], 5)
                while r == 0:
                    c5 += 1
                    M[i][j] = q
                    q, r = divmod(M[i][j], 5)

                M[i][j] = (c2, c5)

        R2 = [[0] * (1 + n) for _ in range(m)]
        R5 = [[0] * (1 + n) for _ in range(m)]
        C2 = [[0] * n for _ in range(1 + m)]
        C5 = [[0] * n for _ in range(1 + m)]

        for i in range(m):
            for j in range(n):
                R2[i][j + 1] = M[i][j][0] + R2[i][j]
                R5[i][j + 1] = M[i][j][1] + R5[i][j]
        for i in range(m):
            for j in range(n):
                C2[i + 1][j] = M[i][j][0] + C2[i][j]
                C5[i + 1][j] = M[i][j][1] + C5[i][j]


        max_path10 = -math.inf

        def combine10(elbow, horiz, verti):
            return min(elbow[0] + verti[0] + horiz[0], elbow[1] + verti[1] + horiz[1])

        # iter each elbow point
        for i in range(m):
            for j in range(n):
                L = (R2[i][j], R5[i][j])
                R = (R2[i][n] - R2[i][j + 1], R5[i][n] - R5[i][j + 1])
                T = (C2[i][j], C5[i][j])
                B = (C2[m][j] - C2[i + 1][j], C5[m][j] - C5[i + 1][j])

                LT10 = combine10(M[i][j], L, T)
                LB10 = combine10(M[i][j], L, B)
                RT10 = combine10(M[i][j], R, T)
                RB10 = combine10(M[i][j], R, B)

                max_path10 = max(max_path10, LT10, LB10, RT10, RB10)

        return max_path10

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