<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:
    bool ConstructSeqHelper(int start, std::vector<bool>& fitted, std::vector<int>& result) {
        bool tried = false;
        for (int i = fitted.size(); i >= 1; --i) {
            if (fitted[i - 1]) {
                continue;
            }
            tried = true;
            if (i > 1) {
                if (start + i >= result.size()) {
                    return false;
                }
                if (result[start + i] > 0) {
                    continue;
                }
                result[start] = i;
                result[start + i] = i;
                fitted[i - 1] = true;
                int next_start = start + 1;
                while (next_start < result.size() && result[next_start] > 0) {
                    ++next_start;
                }
                if (ConstructSeqHelper(next_start, fitted, result)) {
                    return true;
                }
                result[start] = 0;
                result[start + i] = 0;
                fitted[i - 1] = false;
                continue;
            }
            // i == 1
            result[start] = i;
            fitted[i - 1] = true;
            int next_start = start + 1;
            while (next_start < result.size() && result[next_start] > 0) {
                ++next_start;
            }
            if (ConstructSeqHelper(next_start, fitted, result)) {
                return true;
            }
            result[start] = 0;
            fitted[i - 1] = false;
        }
        return !tried;
    }

    vector<int> constructDistancedSequence(int n) {
        std::vector<int> result(2 * n - 1, 0);
        std::vector<bool> fitted(n, false);

        if (!ConstructSeqHelper(0, fitted, result)) {
            return result;
        }
        return result;
    }
};
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