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>

int n,m;
inline int lowbit(int x) {
	return x&(x-1);
}
inline void Pt(int u,int v) {
	printf("%d %d\n",u,v);
}
int main() {
	scanf("%d",&n);
	if(lowbit(n)==0) {
		printf("No\n");
		return 0;
	}
	printf("Yes\n");
	for(int i=1;i<=2;i++)
		Pt(i,i+1),Pt(n+i,n+i+1);
	Pt(3,n+1);
	if(n%2==0)
		m=n--;
	else m=n;
	for(int i=4;i<=n;i++)
		Pt(1,i),Pt(i,m+(i^1));
	if(m>n) {
		int p=lowbit(m);
		Pt(m,p+1),Pt(m-p,2*m);
	}
	return 0;
}