<p>You are given three integers <code>x</code>, <code>y</code>, and <code>z</code>.</p>

<p>You have <code>x</code> strings equal to <code>&quot;AA&quot;</code>, <code>y</code> strings equal to <code>&quot;BB&quot;</code>, and <code>z</code> strings equal to <code>&quot;AB&quot;</code>. You want to choose some (possibly all or none) of these strings and concatenate them in some order to form a new string. This new string must not contain <code>&quot;AAA&quot;</code> or <code>&quot;BBB&quot;</code> as a substring.</p>

<p>Return <em>the maximum possible length of the new string</em>.</p>

<p>A <b>substring</b> is a contiguous <strong>non-empty</strong> sequence of characters within a string.</p>

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

<pre>
<strong>Input:</strong> x = 2, y = 5, z = 1
<strong>Output:</strong> 12
<strong>Explanation: </strong>We can concactenate the strings &quot;BB&quot;, &quot;AA&quot;, &quot;BB&quot;, &quot;AA&quot;, &quot;BB&quot;, and &quot;AB&quot; in that order. Then, our new string is &quot;BBAABBAABBAB&quot;. 
That string has length 12, and we can show that it is impossible to construct a string of longer length.
</pre>

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

<pre>
<strong>Input:</strong> x = 3, y = 2, z = 2
<strong>Output:</strong> 14
<strong>Explanation:</strong> We can concactenate the strings &quot;AB&quot;, &quot;AB&quot;, &quot;AA&quot;, &quot;BB&quot;, &quot;AA&quot;, &quot;BB&quot;, and &quot;AA&quot; in that order. Then, our new string is &quot;ABABAABBAABBAA&quot;. 
That string has length 14, and we can show that it is impossible to construct a string of longer length.
</pre>

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

<ul>
	<li><code>1 &lt;= x, y, z &lt;= 50</code></li>
</ul>

class Solution:
    def longestString(self, x: int, y: int, z: int) -> int:
        
        @cache
        def go(l, x, y, z):
            
            best = 0
            
            if l == "":
                if x > 0:
                    best = max(best, 1 + go("AA", x-1, y, z))
                if y > 0:
                    best = max(best, 1 + go("BB", x, y-1, z))
                if z > 0:
                    best = max(best, 1 + go("AB", x, y, z-1))
            
            elif l == "AA":
                if y > 0:
                    best = max(best, 1 + go("BB", x, y-1, z))
            
            elif l == "BB":
                if x > 0:
                    best = max(best, 1 + go("AA", x-1, y, z))
                if z > 0:
                    best = max(best, 1 + go("AB", x, y, z-1))
                
            elif l == "AB":
                if x > 0:
                    best= max(best, 1 + go("AA", x-1, y, z))
                if z > 0:
                    best = max(best, 1 + go("AB", x, y, z-1))
                
            return best
        
        return 2 * go("", x, y, z)
                
            

                
                
                
        

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