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;
void add(int u, int v) { printf("%d %d\n", u, v); }
int main() {
    scanf("%d", &n);
    if ((n & -n) == n) return puts("No"), 0;
    puts("Yes");
    add(1, 2), add(2, 3), add(3, 1 + n), add(1 + n, 2 + n), add(2 + n, 3 + n);
    for (int i = 4; i + 1 <= n; i += 2)
        add(1, i), add(i, i + 1), add(1, i + 1 + n), add(i + 1 + n, i + n);
    if (~n & 1) add(n, n + n - 1), add(n << 1, n ^ n - 1 ^ 1);
    return 0;
}