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>
using namespace std;
int m,n;
bool ch(int x){while (x!=1){if (x%2)return 0;x/=2;}return 1;}
void sc(int x,int y){cout<<x<<' '<<y<<'\n';}
int lo(int x){return x&(-x);}
int main(){
	cin>>n;
	if (ch(n))puts("No");
	else{
		puts("Yes");
		sc(1,2);sc(2,3);sc(3,n+1);sc(n+1,n+2);sc(n+2,n+3);
		for (int i=4;i<=n;i+=2){
			if (i<n)sc(i,1),sc(i+1,1),sc(i+n,i+1),sc(i+n+1,i);
			else sc(i,i-lo(i)+1),sc(i+n,lo(i));
		}
	}
	return 0;
}