<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:
        
        def check_unique(dict_1, dict_2):
            for k,v in dict_2.items():
                if k in dict_1:
                    return False
            return True
        
        def add(dict_1, dict_2):
            return {**dict_1, **dict_2}
        
        def get_dict(s):
            dict_string = {}
            for el in s:
                if el in dict_string:
                    return {}
                dict_string[el] = 1
        
            return dict_string

        if len(arr) == 1:
            return len(get_dict(arr[0]))

        def backtrack(idx, dict_so_far, curr_len, n):
            if idx >= n:
                return
            nonlocal max_ans

            next_dict = get_dict(arr[idx])
            if check_unique(dict_so_far, next_dict):
                new_dict = add(dict_so_far, next_dict)
                new_len = curr_len + len(next_dict)
                max_ans = max(max_ans, new_len) 

                backtrack(idx + 1, new_dict, new_len, n)
            backtrack(idx + 1, dict_so_far, curr_len, n)
                
        n = len(arr)
        max_ans = 0
        for i in range(n):
            dict_so_far = get_dict(arr[i])
            max_ans = max(max_ans, len(dict_so_far))
            backtrack(i+1, dict_so_far, len(dict_so_far), n)
        return max_ans

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