<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:
        ans = 0
        @cache
        def helper(curr, prev, x, y, z):
            nonlocal ans
            if x == 0 and y == 0 and x == 0:
                ans = max(ans, curr)
                return
            if prev == 'AA' and y == 0:
                ans = max(ans, curr)
                return 
            if prev == 'BB' and x == 0 and z == 0:
                ans = max(ans, curr)
                return
            if prev == 'AB' and x == 0 and z == 0:
                ans = max(ans, curr)
                return
            if prev == 'AA':
                helper(curr+2, 'BB', x, y-1, z)
            if prev == 'BB':
                if x > 0:
                    helper(curr+2, 'AA', x-1, y, z)
                if z > 0:
                    helper(curr+2, 'AB', x, y, z-1)
            if prev == 'AB':
                if x > 0:
                    helper(curr+2, 'AA', x-1, y, z)
                if z > 0:
                    helper(curr+2, 'AB', x, y, z-1)
        if x > 0:
            helper(2, 'AA', x-1, y, z)
        if y > 0:
            helper(2, 'BB', x, y-1, z)
        if z > 0:
            helper(2, 'AB', x, y, z-1)
        return ans

1908