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<cstdio>
#include<algorithm>
using namespace std;
void link(int x,int y){
	printf("%d %d\n",x,y);
}
int main(){
	int n;
	scanf("%d",&n);
	if((1<<__builtin_ctz(n))==n)
		return puts("No"),0;
	puts("Yes");link(1,n+2);
	for(int i=2;i<n;i+=2)
		link(1+n,i),link(i,i+1),link(1+n,i+1+n),link(i+1+n,i+n);
	if(~n&1){
		int m=n&-n;
		link(n,n+m+1);link(n+n,n-m);
	}
	return 0;
}