<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:
        ans = 0
        m, n = len(grid), len(grid[0])
        prefixH = [[[0] * 2 for _ in range(n + 1)] for __ in range(m)]
        prefixV = [[[0] * 2 for _ in range(n)] for __ in range(m + 1)]
        for i in range(m):
            for j in range(n):
                temp= grid[i][j]
                while temp % 2 == 0:
                    prefixH[i][j + 1][0] += 1
                    prefixV[i + 1][j][0] += 1
                    temp //= 2
                while temp % 5 == 0:
                    prefixH[i][j + 1][1] += 1
                    prefixV[i + 1][j][1] += 1
                    temp //= 5
                for k in range(2):
                    prefixH[i][j + 1][k] += prefixH[i][j][k]
                    prefixV[i + 1][j][k] += prefixV[i][j][k]
        for i in range(m):
            for j in range(n):
                left = prefixH[i][j]
                up = prefixV[i][j]
                right, down, center = [0] * 2, [0] * 2, [0] * 2
                for k in range(2):
                    right[k] = prefixH[i][n][k] - prefixH[i][j + 1][k]
                    down[k] = prefixV[m][j][k] - prefixV[i + 1][j][k]
                    center[k] = prefixH[i][j + 1][k] - prefixH[i][j][k]
                LU, LD, RU, RD = [0] * 2, [0] * 2, [0] * 2, [0] * 2
                for k in range(2):
                    LU[k] += left[k] + up[k] + center[k]
                    LD[k] += left[k] + down[k] + center[k]
                    RU[k] += right[k] + up[k] + center[k]
                    RD[k] += right[k] + down[k] + center[k]
                ans = max(ans,
                          min(LU[0], LU[1]),
                          min(LD[0], LD[1]),
                          min(RU[0], RU[1]),
                          min(RD[0], RD[1]))
        return ans
4215