CodeForces - 325E:The Red Button (哈密尔顿 转 欧拉回路)

Piegirl found the red button. You have one last chance to change the inevitable end.

The circuit under the button consists of n nodes, numbered from 0 to n - 1. In order to deactivate the button, the n nodes must be disarmed in a particular order. Node 0 must be disarmed first. After disarming node i, the next node to be disarmed must be either node (2·i) modulo n or node (2·i) + 1 modulo n. The last node to be disarmed must be node 0. Node 0 must be disarmed twice, but all other nodes must be disarmed exactly once.

Your task is to find any such order and print it. If there is no such order, print -1.

Input

Input consists of a single integer n (2 ≤ n ≤ 10⁵).

Output

Print an order in which you can to disarm all nodes. If it is impossible, print -1 instead. If there are multiple orders, print any one of them.

Examples

Input

2

Output

0 1 0

Input

3

Output

-1

Input

4

Output

0 1 3 2 0

Input

16

Output

0 1 2 4 9 3 6 13 10 5 11 7 15 14 12 8 0

题意：给定点[0,N-1]，点i到i*2%N，i*2+1%N连边，问是否存在从0出发的哈密尔顿回路。

思路：N为奇数，无解，因为(N-1)/2要么到N，要么到0，不能满足N和0同时被遍历。

所以考虑N是偶数，发现i和(i+N/2)%N连出去的边相同。 那么发现给个点的入度出度都为2，由于互斥关系，那么入度出度都是1，就是求欧拉回路了； 一定有解，dfs倒序输出，即可。

#include<bits/stdc++.h>
using namespace std;
const int maxn=;
const int Mod=1e9+;
int vis[maxn],ans[maxn],tot,N;
void dfs(int u)
{
    if(vis[u]) return ;
    vis[u]=;
    dfs(u*%N);
    dfs((u*+)%N);
    ans[++tot]=u;
}
int main()
{
    scanf("%d",&N);
    if(N&) return puts("-1"),;
    dfs();
    for(int i=tot;i>=;i--) printf("%d ",ans[i]);
    puts("");
    return ;
}