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 main() {
    ios::sync_with_stdio(false);
    cin.tie(nullptr);
    int n;
    cin >> n;
    if ((n & -n) == n) {
        cout << "No\n";
        return 0;
    }
    cout << "Yes\n";
    for (int i = 1; 2 * i + 1 <= n; ++i) {
        cout << 2 * i << " " << 2 * i + 1 << "\n";
        cout << n + 2 * i << " " << n + 2 * i + 1 << "\n";
        cout << 1 << " " << 2 * i << "\n";
        cout << 1 << " " << n + 2 * i + 1 << "\n";
    }
    cout << 3 << " " << n + 1 << "\n";
    if (n % 2 == 0) {
        int h = 1 << (31 - __builtin_clz(n));
        cout << h << " " << n << "\n";
        cout << 2 * n - h + 1 << " " << 2 * n << "\n";
    }
    return 0;
}