<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:
        
        @lru_cache(None)
        def findBigPalLen(x): #O(n^2)
            if len(x) <= 1: return len(x)
            if x[0] == x[-1]: return 2 + findBigPalLen(x[1:-1])
            return max(findBigPalLen(x[1:]), findBigPalLen(x[:-1]))
        
        @lru_cache(None)
        def findAllPal(x): #Fast pruning
            if len(x) == 0: 
                return [""], [x]
            
            Idx = defaultdict(list)
            for i,l in enumerate(x): 
                Idx[l].append(i)
            
            X, Y = [""], [x]
            for i in range(len(x)):
                X.append(x[i])
                Y.append(x[:i] + x[i + 1:])
                for j in Idx[x[i]]:
                    if j > i:
                        X1, Y1 = findAllPal(x[i + 1 : j])
                        for k in range(len(X1)):
                            X.append(x[i] + X1[k] + x[j])
                            Y.append(x[:i] + Y1[k] + x[j + 1:])
            return X,Y
        X, Y = findAllPal(s)
        return max(len(X[i]) * findBigPalLen(Y[i]) for i in range(len(X)))
19200