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 <bits/stdc++.h>
int n;
int lowbit(int x){
	return (x) &(-x);
}
int main(){
	scanf("%d",&n);
	int t=__builtin_popcount(n);
	if (t==1){
		puts("No");
		return 0;
	} 
	puts("Yes\n1 3\n3 2");
	printf("2 %d\n",1+n);
	printf("%d %d\n",1+n,3+n);
	printf("%d %d\n",n+2,n+3);
	for (int i=4;i<=((n-1)/2)*2+1;i+=2){
		printf("%d 1\n",i+1);
		printf("%d 1\n",i);
		printf("%d %d\n",i,i+1+n);
		printf("%d %d\n",i+n,i+1);
	}
	if (n%2==0){
		int x=lowbit(n),y=n^x;
		printf("%d %d\n",y,n);
		printf("%d %d\n",n+n,x+1);
	}
}