<p>Given an integer <code>n</code>, find a sequence that satisfies all of the following:</p>

<ul>
	<li>The integer <code>1</code> occurs once in the sequence.</li>
	<li>Each integer between <code>2</code> and <code>n</code> occurs twice in the sequence.</li>
	<li>For every integer <code>i</code> between <code>2</code> and <code>n</code>, the <strong>distance</strong> between the two occurrences of <code>i</code> is exactly <code>i</code>.</li>
</ul>

<p>The <strong>distance</strong> between two numbers on the sequence, <code>a[i]</code> and <code>a[j]</code>, is the absolute difference of their indices, <code>|j - i|</code>.</p>

<p>Return <em>the <strong>lexicographically largest</strong> sequence</em><em>. It is guaranteed that under the given constraints, there is always a solution. </em></p>

<p>A sequence <code>a</code> is lexicographically larger than a sequence <code>b</code> (of the same length) if in the first position where <code>a</code> and <code>b</code> differ, sequence <code>a</code> has a number greater than the corresponding number in <code>b</code>. For example, <code>[0,1,9,0]</code> is lexicographically larger than <code>[0,1,5,6]</code> because the first position they differ is at the third number, and <code>9</code> is greater than <code>5</code>.</p>

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

<pre>
<strong>Input:</strong> n = 3
<strong>Output:</strong> [3,1,2,3,2]
<strong>Explanation:</strong> [2,3,2,1,3] is also a valid sequence, but [3,1,2,3,2] is the lexicographically largest valid sequence.
</pre>

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

<pre>
<strong>Input:</strong> n = 5
<strong>Output:</strong> [5,3,1,4,3,5,2,4,2]
</pre>

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

<ul>
	<li><code>1 &lt;= n &lt;= 20</code></li>
</ul>

class Solution {
public:
    vector<int> sol{};

    bool helper(unordered_set<int>& set, int idx, vector<int>& current, int n)
    {
        if(idx >= current.size())
        {
            sol = current;
            return 1;
        }
        
        if(current[idx] != -1) return helper(set, idx + 1, current, n);
        else{

            for(int i = n; i >= 1 && sol.empty(); i--)
            {
                if(set.contains(i)) continue;
                set.insert(i);
                if(i == 1)
                {
                    current[idx] = 1;
                    if(helper(set, idx + 1, current, n)) return 1;
                    current[idx] = -1;
                }
                else if(idx + i < current.size() && current[idx + i] == -1)
                {
                    current[idx] = i;
                    current[idx + i] = i;
                    if(helper(set, idx + 1, current, n)) return 1;
                    current[idx] = -1;
                    current[idx + i] = -1;
                }
                set.erase(i);
            }
        }
        return 0;
    }

    vector<int> constructDistancedSequence(int n) {
        unordered_set<int> set{};
        vector<int> current(2*n - 1, -1);
        helper(set, 0, current, n);
        return sol;
    }
};
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