<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:
        # Set to store unique characters from the current combination
        unique_chars = set()

        def hasDuplicates(existing_chars: set, new_string: str) -> bool:
            """
            Check if adding characters from new_string to existing_chars results in duplicates.
            """
            combined_counter = Counter(existing_chars) + Counter(new_string)
            return any(count > 1 for count in combined_counter.values())

        def backtrack(index: int) -> int:
            """
            Use backtracking to explore all possible combinations.
            """
            # Base case: if we've considered all strings in arr
            if index == len(arr):
                return len(unique_chars)

            # Option 1: Exclude the current string and proceed
            max_length = backtrack(index + 1)

            # Option 2: Include the current string if it doesn't introduce duplicates
            if not hasDuplicates(unique_chars, arr[index]):
                # Add current string's characters to the unique set
                unique_chars.update(arr[index])
                # Recursively calculate the max length by including this string
                max_length = max(max_length, backtrack(index + 1))
                # Backtrack: remove current string's characters after processing
                unique_chars.difference_update(arr[index])

            return max_length

        # Start backtracking from the first index
        return backtrack(0)

446