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 main(){
    using namespace std;
    size_t N;
    cin >> N;
    if(__builtin_popcountll(N) == 1)return 0 & puts("No");
    puts("Yes\n1 2");
    for(unsigned long i = 2; i < N; i += 2)cout << i << " " << i + 1 << "\n" << i + 1 << " " << N + 1 << "\n" << N + 1 << " " << N + i << "\n" << N + i << " " << N + i + 1 << endl;
    if(N % 2 == 0)cout << N << " " << N + (N & -N) << "\n" << ((N & -N) ^ 1 ^ N) << " " << 2 * N << endl;
    return 0;
}