<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>

class Solution:
    def maxLength(self, arr: List[str]) -> int:
        # intuition:
        # we want to create all combinations of the strings in the array
        # only continue down the branch of the string does not contain duplicate chars
        # if duplicate, backtrack and move on
        # continue until we reach end of list
        # every time we get a new valid substring, compare length with res and take max

        # Essentially, we either take it or we don't, but if duplicate char we backtrack
         

        def has_duplicate(word):
            return len(set(word)) != len(word)

        res = 0
        charcount = defaultdict(int)

        def dfs(i, curlength):
            nonlocal res

            # Check if any character has appeared more than once
            if any(count > 1 for count in charcount.values()):
                return

            # Update result with the maximum length found
            res = max(res, curlength)

            # Base case: if we've processed all elements
            if i == len(arr):
                return

            for j in range(i, len(arr)):
                wordToadd = arr[j]

                # Skip the word if it contains duplicate characters internally
                if has_duplicate(wordToadd):
                    continue

                # Add current word's characters to the character count
                for letter in wordToadd:
                    charcount[letter] += 1

                # Explore the next element with the current word added
                dfs(j + 1, curlength + len(wordToadd))

                # Backtrack by removing the characters of the current word
                for letter in wordToadd:
                    charcount[letter] -= 1

        # Start DFS with the first word and length 0
        dfs(0, 0)
        return res

149