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>

void pt(int a,int b)
{
  printf("%d %d\n",a,b);
}

int main()
{
  int n;
  scanf("%d",&n);
  int p=1;
  while(n%(2*p)==0){
    p*=2;
  }
  if(n==p){
    puts("No");
    return 0;
  }
  puts("Yes");
  for(int i=0;i<(n-1)/2;i++){
    pt(1,2*(i+1));
    pt(2*(i+1),2*(i+1)+1);
    pt(1,2*(i+1)+n+1);
    pt(2*(i+1)+n+1,2*(i+1)+n);
  }
  pt(n+1,3);
  if(n%2==0){
    pt(p+1+n,n);
    pt(n-p,2*n);
  }
  return 0;
}