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 n;

int main(){
    cin>>n;
    if (n==(n&-n)) return puts("No"), 0;
    puts("Yes\n1 2\n2 3");
    printf("3 %d\n%d %d\n%d %d\n",n+1,n+1,n+2,n+2,n+3);
    for (int i=4;i<n;i+=2){
        printf("%d %d\n%d 1\n1 %d\n%d %d\n",i+1,i,i,i+1+n,i+1+n,i+n);
    }
    if (n%2==0){
        for (int i=20;i>=0;--i)
            if (n>>i&1){
                printf("%d %d\n%d %d\n",n,n-(1<<i),(1<<i)+1+n,n*2);
                break;
            }
    }
}