<p>You are given a <strong>0-indexed</strong> array of <strong>unique</strong> strings <code>words</code>.</p>

<p>A <strong>palindrome pair</strong> is a pair of integers <code>(i, j)</code> such that:</p>

<ul>
	<li><code>0 &lt;= i, j &lt; words.length</code>,</li>
	<li><code>i != j</code>, and</li>
	<li><code>words[i] + words[j]</code> (the concatenation of the two strings) is a <span data-keyword="palindrome-string">palindrome</span>.</li>
</ul>

<p>Return <em>an array of all the <strong>palindrome pairs</strong> of </em><code>words</code>.</p>

<p>You must write an algorithm with&nbsp;<code>O(sum of words[i].length)</code>&nbsp;runtime complexity.</p>

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

<pre>
<strong>Input:</strong> words = [&quot;abcd&quot;,&quot;dcba&quot;,&quot;lls&quot;,&quot;s&quot;,&quot;sssll&quot;]
<strong>Output:</strong> [[0,1],[1,0],[3,2],[2,4]]
<strong>Explanation:</strong> The palindromes are [&quot;abcddcba&quot;,&quot;dcbaabcd&quot;,&quot;slls&quot;,&quot;llssssll&quot;]
</pre>

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

<pre>
<strong>Input:</strong> words = [&quot;bat&quot;,&quot;tab&quot;,&quot;cat&quot;]
<strong>Output:</strong> [[0,1],[1,0]]
<strong>Explanation:</strong> The palindromes are [&quot;battab&quot;,&quot;tabbat&quot;]
</pre>

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

<pre>
<strong>Input:</strong> words = [&quot;a&quot;,&quot;&quot;]
<strong>Output:</strong> [[0,1],[1,0]]
<strong>Explanation:</strong> The palindromes are [&quot;a&quot;,&quot;a&quot;]
</pre>

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

<ul>
	<li><code>1 &lt;= words.length &lt;= 5000</code></li>
	<li><code>0 &lt;= words[i].length &lt;= 300</code></li>
	<li><code>words[i]</code> consists of lowercase English letters.</li>
</ul>

from functools import cache

class Solution:
    @cache
    def checkPalin(self, word, i, j):
        if i > j:
            return True
        if word[i] != word[j]:
            return False
        return self.checkPalin(word, i+1, j-1)

    def palindromePairs(self, words: List[str]) -> List[List[int]]:
        backwards = dict()
        output = []
        for i, word in enumerate(words):
            backwards[word[::-1]] = i
        
        for i, word in enumerate(words):
            n = len(word)
            if word in backwards and backwards[word] != i:
                output.append([i, backwards[word]])
            
            if "" in backwards and word != "" and self.checkPalin(word, 0, n-1):
                output.append([backwards[""], i])
                output.append([i, backwards[""]])
            
            for j in range(1, n):
                left = word[:j]
                if left in backwards and backwards[left] != i and self.checkPalin(word, j, n-1):
                    output.append([i, backwards[left]])
                right = word[j:]
                if right in backwards and backwards[right] != i and self.checkPalin(word, 0, j-1):
                    output.append([backwards[right], i])

        return output
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