<p>Given a string <code>s</code>, find two <strong>disjoint palindromic subsequences</strong> of <code>s</code> such that the <strong>product</strong> of their lengths is <strong>maximized</strong>. The two subsequences are <strong>disjoint</strong> if they do not both pick a character at the same index.</p>

<p>Return <em>the <strong>maximum</strong> possible <strong>product</strong> of the lengths of the two palindromic subsequences</em>.</p>

<p>A <strong>subsequence</strong> is a string that can be derived from another string by deleting some or no characters without changing the order of the remaining characters. A string is <strong>palindromic</strong> if it reads the same forward and backward.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="example-1" src="https://assets.leetcode.com/uploads/2021/08/24/two-palindromic-subsequences.png" style="width: 550px; height: 124px;" />
<pre>
<strong>Input:</strong> s = &quot;leetcodecom&quot;
<strong>Output:</strong> 9
<strong>Explanation</strong>: An optimal solution is to choose &quot;ete&quot; for the 1<sup>st</sup> subsequence and &quot;cdc&quot; for the 2<sup>nd</sup> subsequence.
The product of their lengths is: 3 * 3 = 9.
</pre>

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

<pre>
<strong>Input:</strong> s = &quot;bb&quot;
<strong>Output:</strong> 1
<strong>Explanation</strong>: An optimal solution is to choose &quot;b&quot; (the first character) for the 1<sup>st</sup> subsequence and &quot;b&quot; (the second character) for the 2<sup>nd</sup> subsequence.
The product of their lengths is: 1 * 1 = 1.
</pre>

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

<pre>
<strong>Input:</strong> s = &quot;accbcaxxcxx&quot;
<strong>Output:</strong> 25
<strong>Explanation</strong>: An optimal solution is to choose &quot;accca&quot; for the 1<sup>st</sup> subsequence and &quot;xxcxx&quot; for the 2<sup>nd</sup> subsequence.
The product of their lengths is: 5 * 5 = 25.
</pre>

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

<ul>
	<li><code>2 &lt;= s.length &lt;= 12</code></li>
	<li><code>s</code> consists of lowercase English letters only.</li>
</ul>

class Solution:
    def maxProduct(self, s: str) -> int:
        n = len(s)
        palindrome_info = []  # List to store (bitmask, length) for each palindrome

        # Helper function to check if a subsequence (represented by bitmask) is a palindrome
        def is_palindrome(mask):
            subseq = [s[i] for i in range(n) if mask & (1 << i)]
            return subseq == subseq[::-1]

        # Backtrack to generate all subsequences and check for palindromes
        def backtrack(start, path_mask):
            if is_palindrome(path_mask):
                palindrome_info.append((path_mask, bin(path_mask).count('1')))  # Store bitmask and its length

            for i in range(start, n):
                backtrack(i + 1, path_mask | (1 << i))  # Add current character to bitmask and recurse

        backtrack(0, 0)  # Start backtracking from index 0 with empty bitmask

        # Now find two disjoint palindromic subsequences with the maximum product of lengths
        max_product = 0
        m = len(palindrome_info)

        for i in range(m):
            mask1, len1 = palindrome_info[i]
            for j in range(i + 1, m):
                mask2, len2 = palindrome_info[j]
                # Check if the two palindromes are disjoint using bitwise AND
                if mask1 & mask2 == 0:
                    max_product = max(max_product, len1 * len2)

        return max_product
17800