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 JAVA):

Solution:
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Scanner;

class Main {
	public static void main(String[] args) {
		new Main().run();
	}

	void run() {
		Scanner sc = new Scanner(System.in);
		int N = sc.nextInt();
		{
			for (long i = 1; i < 1L << 30; i *= 2) {
				if (i == N) {
					System.out.println("No");
					return;
				}
			}
		}

		System.out.println("Yes");
		int M = (N % 2 == 0 ? N - 1 : N);
		for (int i = 2; i <= M; ++i) {
			System.out.println(1 + " " + i);
		}
		for (int i = 2; i <= M; i += 2) {
			System.out.println((i + 1) + " " + (i + N));
		}
		for (int i = 3; i <= M; i += 2) {
			System.out.println((i - 1) + " " + (i + N));
		}
		System.out.println((2 + N) + " " + (1 + N));
		if (M != N) {
			for (int i = 2; i <= M; ++i) {
				if (2 <= (1 ^ i ^ N) && (1 ^ i ^ N) <= M && (1 ^ i ^ N) != i) {
					System.out.println(i + " " + N);
					System.out.println((1 ^ i ^ N) + " " + (N + N));
					return;
				}
			}

		}
	}

	void tr(Object... objects) {
		System.out.println(Arrays.deepToString(objects));
	}
}