<p>You are given an array of strings <code>arr</code>. A string <code>s</code> is formed by the <strong>concatenation</strong> of a <strong>subsequence</strong> of <code>arr</code> that has <strong>unique characters</strong>.</p>

<p>Return <em>the <strong>maximum</strong> possible length</em> of <code>s</code>.</p>

<p>A <strong>subsequence</strong> is an array that can be derived from another array by deleting some or no elements without changing the order of the remaining elements.</p>

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

<pre>
<strong>Input:</strong> arr = [&quot;un&quot;,&quot;iq&quot;,&quot;ue&quot;]
<strong>Output:</strong> 4
<strong>Explanation:</strong> All the valid concatenations are:
- &quot;&quot;
- &quot;un&quot;
- &quot;iq&quot;
- &quot;ue&quot;
- &quot;uniq&quot; (&quot;un&quot; + &quot;iq&quot;)
- &quot;ique&quot; (&quot;iq&quot; + &quot;ue&quot;)
Maximum length is 4.
</pre>

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

<pre>
<strong>Input:</strong> arr = [&quot;cha&quot;,&quot;r&quot;,&quot;act&quot;,&quot;ers&quot;]
<strong>Output:</strong> 6
<strong>Explanation:</strong> Possible longest valid concatenations are &quot;chaers&quot; (&quot;cha&quot; + &quot;ers&quot;) and &quot;acters&quot; (&quot;act&quot; + &quot;ers&quot;).
</pre>

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

<pre>
<strong>Input:</strong> arr = [&quot;abcdefghijklmnopqrstuvwxyz&quot;]
<strong>Output:</strong> 26
<strong>Explanation:</strong> The only string in arr has all 26 characters.
</pre>

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

<ul>
	<li><code>1 &lt;= arr.length &lt;= 16</code></li>
	<li><code>1 &lt;= arr[i].length &lt;= 26</code></li>
	<li><code>arr[i]</code> contains only lowercase English letters.</li>
</ul>

from typing import List
from collections import Counter

class Solution:
    def maxLength(self, arr: List[str]) -> int:
        # Initialize an empty set to keep track of unique characters
        charSet = set()

        # Helper function to check if there is any overlap between charSet and string s
        def overlap(charSet, s):
            # Count characters in charSet and s, and check if any character count is greater than 1
            c = Counter(charSet) + Counter(s)
            return max(c.values()) > 1
            
            # Alternatively, manually check for overlap without using Counter
            # prev = set()
            # for c in s:
            #     if c in charSet or c in prev:
            #         return True
            #     prev.add(c)
            # return False

        # Backtracking function to find the maximum length of concatenated string with unique characters
        def backtrack(i):
            # Base case: if we've considered all strings, return the length of the unique character set
            if i == len(arr):
                return len(charSet)
            
            res = 0
            # Check if adding arr[i] would create any overlap
            if not overlap(charSet, arr[i]):
                # Add characters of arr[i] to charSet
                for c in arr[i]:
                    charSet.add(c)
                # Recurse with the next string in the list
                res = backtrack(i + 1)
                # Remove characters of arr[i] from charSet to backtrack
                for c in arr[i]:
                    charSet.remove(c)
            # Compare the result of including arr[i] and not including arr[i]
            return max(res, backtrack(i + 1))

        # Start the backtracking process from the first string
        return backtrack(0)

378