<p>Given two arrays <code>nums1</code>&nbsp;and <code><font face="monospace">nums2</font></code><font face="monospace">.</font></p>

<p>Return the maximum dot product&nbsp;between&nbsp;<strong>non-empty</strong> subsequences of nums1 and nums2 with the same length.</p>

<p>A subsequence of a array is a new array which is formed from the original array by deleting some (can be none) of the characters without disturbing the relative positions of the remaining characters. (ie,&nbsp;<code>[2,3,5]</code>&nbsp;is a subsequence of&nbsp;<code>[1,2,3,4,5]</code>&nbsp;while <code>[1,5,3]</code>&nbsp;is not).</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> nums1 = [2,1,-2,5], nums2 = [3,0,-6]
<strong>Output:</strong> 18
<strong>Explanation:</strong> Take subsequence [2,-2] from nums1 and subsequence [3,-6] from nums2.
Their dot product is (2*3 + (-2)*(-6)) = 18.</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> nums1 = [3,-2], nums2 = [2,-6,7]
<strong>Output:</strong> 21
<strong>Explanation:</strong> Take subsequence [3] from nums1 and subsequence [7] from nums2.
Their dot product is (3*7) = 21.</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> nums1 = [-1,-1], nums2 = [1,1]
<strong>Output:</strong> -1
<strong>Explanation: </strong>Take subsequence [-1] from nums1 and subsequence [1] from nums2.
Their dot product is -1.</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= nums1.length, nums2.length &lt;= 500</code></li>
	<li><code>-1000 &lt;= nums1[i], nums2[i] &lt;= 1000</code></li>
</ul>

class Solution:
    def maxDotProduct(self, nums1: List[int], nums2: List[int]) -> int:
        MIN = -1000 * 1000
        m,n = len(nums1), len(nums2)
        dp = [[MIN]*(n+1) for i in range(m+1)]
        def dfs(i,j):
            if i == m or j == n:
                return 0
            if dp[i][j] != MIN:
                return dp[i][j]
            
            opt1 = nums1[i]*nums2[j] + dfs(i+1, j+1)
            opt2 = dfs(i+1, j)
            opt3 = dfs(i, j+1)
            
            res = max(opt1, opt2, opt3)
            dp[i][j] = res
            return res
        
        first_max = float('-inf')
        second_max = float('-inf')
        first_min = float('inf')
        second_min = float('inf')

        # Calculate maximum and minimum values for nums1
        for num in nums1:
            first_max = max(first_max, num)
            first_min = min(first_min, num)
        # Calculate maximum and minimum values for nums2
        for num in nums2:
            second_max = max(second_max, num)
            second_min = min(second_min, num)

        # Check special cases where all elements are negative
        if (first_max < 0 and second_min > 0) or (first_min > 0 and second_max < 0):
            return max(first_max * second_min, first_min * second_max)
        
        
        return dfs(0,0)

            

            
        
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