<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 {
private:
    int n;
    std::vector<int> sequence;
    std::unordered_set<int> nums;
    bool constructSequence(int i) {
        if (i >= sequence.size()) {
            return nums.size() == 0;
        } else if (sequence[i] != 0) {
            return constructSequence(i + 1);
        }

        for (int num = n; num > 1; num--) {
            if (nums.contains(num) && i + num < sequence.size() && sequence[i + num] == 0) {
                nums.erase(num);
                sequence[i] = num;
                sequence[i + num] = num;
                if (constructSequence(i + 1)) {
                    return true;
                } else {
                    nums.insert(num);
                    sequence[i] = 0;
                    sequence[i + num] = 0;
                }
            }
        }

        if (nums.contains(1)) {
            nums.erase(1);
            sequence[i] = 1;
            if (constructSequence(i + 1)) {
                return true;
            } else {
                nums.insert(1);
                sequence[i] = 0;
            }
        }

        return false;
    }

public:
    vector<int> constructDistancedSequence(int n) {
        /**
        5,0,0,0,5,0,0,0,0
        */
        this->n = n;
        for (int num = n; num > 0; num--) {
            nums.insert(num);
        }
        for (int i = 0; i < 2 * n - 1; i++) {
            sequence.push_back(0);
        }

        constructSequence(0);
        return sequence;
    }
};
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