<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 {
public:
    int maxScore(vector<int>& nums) {
        function<int(int&,int&)> gcd_ = [&](int& a, int& b)->int {
            if(a > b) return gcd_(b, a);
            if(a == 0) return b;
            int c = b % a;
            return gcd_(c, a);
        };
        int n = nums.size();

        vector<vector<int>> gcd(n, vector<int>(n));
        for(int i = 0; i < n; i++) {
            for(int j = 0; j < n; j++) {
                gcd[i][j] = gcd_(nums[i], nums[j]);
            }
        }

        int ans = 0;
        vector<int> g(n/2);
        int sz = 0;
        int visited;
        function<void()> bt = [&]() {
            if(sz == n/2){
                vector<int> gc(g);
                sort(gc.begin(), gc.end());
                int cur = 0;
                for(int i = 0; i < n/2; i++) {
                    cur += (i+1)*gc[i];
                }
                ans = max(ans, cur);
                return;
            } 

            int id = 0;
            while((visited & (1<<id)) != 0) id++;
            
            for(int i = id+1; i < n; i++) {
                if((visited & (1<<i)) == 0) {
                    g[sz++] = gcd[id][i];
                    visited = visited ^ (1<<id) ^ (1<<i);
                    bt();
                    visited = visited ^ (1<<id) ^ (1<<i);
                    sz--;
                }
            }
        };
        bt();
        return ans;
    }
};
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