<p>You are given <code>nums</code>, an array of positive integers of size <code>2 * n</code>. You must perform <code>n</code> operations on this array.</p>

<p>In the <code>i<sup>th</sup></code> operation <strong>(1-indexed)</strong>, you will:</p>

<ul>
	<li>Choose two elements, <code>x</code> and <code>y</code>.</li>
	<li>Receive a score of <code>i * gcd(x, y)</code>.</li>
	<li>Remove <code>x</code> and <code>y</code> from <code>nums</code>.</li>
</ul>

<p>Return <em>the maximum score you can receive after performing </em><code>n</code><em> operations.</em></p>

<p>The function <code>gcd(x, y)</code> is the greatest common divisor of <code>x</code> and <code>y</code>.</p>

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

<pre>
<strong>Input:</strong> nums = [1,2]
<strong>Output:</strong> 1
<strong>Explanation:</strong>&nbsp;The optimal choice of operations is:
(1 * gcd(1, 2)) = 1
</pre>

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

<pre>
<strong>Input:</strong> nums = [3,4,6,8]
<strong>Output:</strong> 11
<strong>Explanation:</strong>&nbsp;The optimal choice of operations is:
(1 * gcd(3, 6)) + (2 * gcd(4, 8)) = 3 + 8 = 11
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> nums = [1,2,3,4,5,6]
<strong>Output:</strong> 14
<strong>Explanation:</strong>&nbsp;The optimal choice of operations is:
(1 * gcd(1, 5)) + (2 * gcd(2, 4)) + (3 * gcd(3, 6)) = 1 + 4 + 9 = 14
</pre>

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

<ul>
	<li><code>1 &lt;= n &lt;= 7</code></li>
	<li><code>nums.length == 2 * n</code></li>
	<li><code>1 &lt;= nums[i] &lt;= 10<sup>6</sup></code></li>
</ul>

class Solution:
    def maxScore(self, nums: List[int]) -> int:
        n = len(nums) // 2
        num2mask = {i:(1<<i) for i in range(len(nums))}
        def gcd(a,b):
            a,b = max(a,b), min(a,b)
            while a % b > 0:
                a,b = b,a%b
            return b
        gcd_mat = [[0] * len(nums) for i in range(len(nums))]
        for i in range(len(nums)):
            for j in range(i+1, len(nums)):
                gcd_mat[i][j] = gcd(nums[i], nums[j])
        for i in range(len(nums)):
            for j in range(i):
                gcd_mat[i][j] = gcd_mat[j][i]
        maxScore = 0
        def backtrack(scores, i, mask, n):
            if len(scores) == n:
                scores = sorted(scores)
                nonlocal maxScore
                newScore = 0
                for i in range(1, n+1):
                    newScore += (i * scores[i-1])
                maxScore = max(maxScore, newScore)
                return
            imask = num2mask[i]
            if imask & mask:
                backtrack(scores, i+1, mask, n)
                return
            mask |= imask
            for j in range(i+1, len(nums)):
                jmask = num2mask[j]
                if jmask & mask:
                    continue
                scores.append(gcd_mat[i][j])
                backtrack(scores, i+1, mask|jmask, n)
                scores.pop()
        scores = []
        backtrack(scores, 0, 0, n)
        return maxScore

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