<p>You are given two strings, <code>word1</code> and <code>word2</code>. You want to construct a string in the following manner:</p>

<ul>
	<li>Choose some <strong>non-empty</strong> subsequence <code>subsequence1</code> from <code>word1</code>.</li>
	<li>Choose some <strong>non-empty</strong> subsequence <code>subsequence2</code> from <code>word2</code>.</li>
	<li>Concatenate the subsequences: <code>subsequence1 + subsequence2</code>, to make the string.</li>
</ul>

<p>Return <em>the <strong>length</strong> of the longest <strong>palindrome</strong> that can be constructed in the described manner. </em>If no palindromes can be constructed, return <code>0</code>.</p>

<p>A <strong>subsequence</strong> of a string <code>s</code> is a string that can be made by deleting some (possibly none) characters from <code>s</code> without changing the order of the remaining characters.</p>

<p>A <strong>palindrome</strong> is a string that reads the same forward&nbsp;as well as backward.</p>

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

<pre>
<strong>Input:</strong> word1 = &quot;cacb&quot;, word2 = &quot;cbba&quot;
<strong>Output:</strong> 5
<strong>Explanation:</strong> Choose &quot;ab&quot; from word1 and &quot;cba&quot; from word2 to make &quot;abcba&quot;, which is a palindrome.</pre>

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

<pre>
<strong>Input:</strong> word1 = &quot;ab&quot;, word2 = &quot;ab&quot;
<strong>Output:</strong> 3
<strong>Explanation:</strong> Choose &quot;ab&quot; from word1 and &quot;a&quot; from word2 to make &quot;aba&quot;, which is a palindrome.</pre>

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

<pre>
<strong>Input:</strong> word1 = &quot;aa&quot;, word2 = &quot;bb&quot;
<strong>Output:</strong> 0
<strong>Explanation:</strong> You cannot construct a palindrome from the described method, so return 0.</pre>

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

<ul>
	<li><code>1 &lt;= word1.length, word2.length &lt;= 1000</code></li>
	<li><code>word1</code> and <code>word2</code> consist of lowercase English letters.</li>
</ul>

class Solution:
    def longestPalindrome(self, word1: str, word2: str) -> int:
        word3 = word1 + word2
        len_word1 = len(word1)
        len_word2 = len(word2)
        len_word3 = len(word3)
        
        # Initialize memoization table with None
        memo = [[[None for _ in range(2)] for _ in range(len_word3)] for _ in range(len_word3)]
        
        def dfs(i, j, taken):
            if i > j:
                if taken:
                    return 0
                return -float('inf')
            if i == j:
                if taken:
                    return 1
                return -float('inf')
            
            if memo[i][j][taken] is not None:
                return memo[i][j][taken]
            
            if word3[i] == word3[j]:
                if i < len_word1 and j >= len_word1:
                    memo[i][j][taken] = dfs(i+1, j-1, True) + 2
                else:
                    memo[i][j][taken] = dfs(i+1, j-1, taken) + 2
            else:
                memo[i][j][taken] = max(dfs(i+1, j, taken), dfs(i, j-1, taken), dfs(i+1, j-1, taken))
            
            return memo[i][j][taken]
        
        result = dfs(0, len_word3 - 1, 0)
        return result if result != -float('inf') else 0        
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