Given a positive integer $n$, construct a permutation $a_0, \ldots, a_{n-1}$ of $0 \sim n-1$ such that the sequence $b_0, \ldots, b_{n-1}$ defined by $b_i := |a_i - i|$ is also a permutation of $0 \sim n-1$. If no such permutation exists, determine that it is impossible.
Input
A single line containing a positive integer $n$.
Output
If no solution exists, output a single line containing the string NO.
Otherwise, output YES on the first line, and on the second line, output $n$ non-negative integers $a_0, \ldots, a_{n-1}$ representing the constructed permutation $a$.
If there are multiple solutions, you may output any one of them.
Examples
Input 1
1
Output 1
YES 0
Input 2
3
Output 2
NO
Input 3
4
Output 3
YES 3 0 2 1
Constraints
There are 25 test cases in total, each worth 4 points. For all test cases, $n \le 10^6$.
- For test cases $1 \sim 7$, $n$ are $114514, 206669, 265720, 324765, 620012, 797161, 974304$ respectively.
- For test cases $8 \sim 10$, $n \equiv 0 \pmod{12}$.
- For test cases $11 \sim 13$, $n \equiv 1 \pmod{12}$.
- For test cases $14 \sim 16$, $n \equiv 4 \pmod{12}$.
- For test cases $17 \sim 19$, $n \equiv 5 \pmod{12}$.
- For test cases $20 \sim 22$, $n \equiv 8 \pmod{12}$.
- For test cases $23 \sim 25$, $n \equiv 9 \pmod{12}$.