<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:

        mem = {}

        def helper(aa, bb, ab, last):
            key = (aa, bb, ab, last)
            if key in mem:
                return mem[key]
            
            result = 0
            if last == "aa":
                if bb == 0:
                    result = 2
                else:
                    result = 2 + helper(aa, bb-1, ab, "bb")
            else:
                r1 = 2
                if aa > 0:
                    r1 += helper(aa-1, bb, ab, "aa")
                r2 = 2
                if ab > 0:
                    r2 += helper(aa, bb, ab-1, "ab")
                result = max(r1, r2)

            mem[key] = result
            return result
        
        res = 0
        if x > 0:
            res = max(res, helper(x-1, y, z, "aa"))
        if y > 0:
            res = max(res, helper(x, y-1, z, "bb"))
        if z > 0:
            res = max(res, helper(x, y, z-1, "ab"))
        
        return res
2119