<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:

        # AA = 0, BB = 1, AB = 2
        adj_list = {
            0: [1],
            1: [2, 0],
            2: [2, 0]
        }

        lst = [x,y, z]
        memo = {}
        def recursive(lst, cur):
            nonlocal memo
            tup = (cur, lst[0], lst[1], lst[2])
            if(tup in memo):
                return memo[tup]

            if(sum(lst) == 0):
                return 0
            
            if(lst[cur] == 0):
                return 0
            
            lst[cur] -=1
            cur_max = 0
            for item in adj_list[cur]:

                cur_max = max(cur_max,  recursive(lst, item))
            
            lst[cur] +=1

            memo[tup] = 2 + cur_max
            return memo[tup]
        
        return max(recursive(lst, 0), recursive(lst, 1), recursive(lst, 2))




        

        
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