Problem: You are given an integer N. Determine if there exists a tree with 2N vertices numbered 1 to 2N satisfying the following condition, and show one such tree if the answer is yes.

* Assume that, for each integer i between 1 and N (inclusive), Vertex i and N+i have the weight i. Then, for each integer i between 1 and N, the bitwise XOR of the weights of the vertices on the path between Vertex i and N+i (including themselves) is i.

Constraints

* N is an integer.
* 1 \leq N \leq 10^{5}

Input

Input is given from Standard Input in the following format:


N


Output

If there exists a tree satisfying the condition in the statement, print `Yes`; otherwise, print `No`. Then, if such a tree exists, print the 2N-1 edges of such a tree in the subsequent 2N-1 lines, in the following format:


a_{1} b_{1}
\vdots
a_{2N-1} b_{2N-1}


Here each pair (a_i, b_i) means that there is an edge connecting Vertex a_i and b_i. The edges may be printed in any order.

Examples

Input

3


Output

Yes
1 2
2 3
3 4
4 5
5 6


Input

1


Output

No

Solution (in CPP):

Solution:
#include<stdio.h>
#include<iostream>
#define out(u,v) printf("%d %d\n",u,v)
#define lowbit(x) (x&(-x))
using namespace std;
int n,m,a,b;
int main(){
    // freopen("atcoder5140.in","r",stdin);
    // freopen("atcoder5140.out","w",stdout);
    scanf("%d",&n);
    if(n==lowbit(n)||n<3) return puts("No"),0;
    puts("Yes");
    out(1,2);out(2,3);out(3,n+1);out(n+1,n+2);out(n+2,n+3);
    for(int i=4;i<n;i+=2) out(1,i+1),out(i+1,i),out(1,i+n),out(i+n,i+1+n);
    if(!(n&1)) out(n,lowbit(n)+n),out(n+n,(n^lowbit(n))^1);
    return 0;
}