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 PYTHON3):

Solution:
import sys, collections

def solve():
    input = sys.stdin.readline
    N = int(input())
    k = 1
    while k <= N:
        if k == N:
            print("No")
            break
        else: k *= 2
    else:
        if N % 2 == 1:
            print("Yes")
            ans = ""
            ans += "1 2\n"
            ans += "2 3\n"
            ans += "3 " + str(N + 1) + "\n"
            ans += str(N + 1) + " " + str(N + 2) + "\n"
            ans += str(N + 2) + " " + str(N + 3) + "\n"
            for i in range(2, N // 2 + 1):
                ans += str(2 * i) + " " + str(2 * i + 1) + "\n"
                ans += str(2*i+1) + " 1\n"
                ans += "1 " + str(N + 2*i) + "\n"
                ans += str(N+2*i) + " " + str(N+2*i+1) + "\n"
            print(ans)
        else:
            print("Yes")
            ans = ""
            ans += "1 2\n"
            ans += "2 3\n"
            ans += "3 " + str(N + 1) + "\n"
            ans += str(N + 1) + " " + str(N + 2) + "\n"
            ans += str(N + 2) + " " + str(N + 3) + "\n"
            for i in range(2, N // 2):
                ans += str(2 * i) + " " + str(2 * i + 1) + "\n"
                ans += str(2*i+1) + " 1\n"
                ans += "1 " + str(N + 2*i) + "\n"
                ans += str(N+2*i) + " " + str(N+2*i+1) + "\n"
            b = N ^ (N - 1)
            ans += str(N) + " " + str(N-1) + "\n"
            if b % 2 == 1:
                ans += str(N + b - 1) + " " + str(2 * N)
            else:
                ans += str(b + 1) + " " + str(2 * N)
            print(ans)

    return 0

if __name__ == "__main__":
    solve()