<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:
        converted = []
        for s in arr:
            val = 0
            for char in s:
                bit = self.char_to_bit(char)
                if bit & val:
                    val = 0
                    break
                val = bit | val
            if val != 0:
                converted.append(val)
        n = len(converted)
        best_result = 0
        path = [(0, 0)]
        while len(path) > 0:
            next_i, cur_num = heapq.heappop(path)
            if next_i == n:
                result = self.bit_count(cur_num)
                if result > best_result:
                    best_result = result
                continue
            if not (cur_num & converted[next_i]):
                heapq.heappush(path, (next_i+1, cur_num | converted[next_i]))
            heapq.heappush(path, (next_i+1, cur_num))
        return best_result
        
    def bit_count(self, num):
        result = 0
        while num > 0:
            result += num % 2
            num //= 2
        return result

    def char_to_bit(self, char):
        raw = ord(char) - ord('a')
        return 2**raw
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