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:
#!/usr/bin/env python3
import sys

YES = "Yes"  # type: str
NO = "No"  # type: str

def solve(N: int):
    k = 1
    while k < N:
        k *= 2
    if N == k:
        print(NO)
        return
    print(YES)

    if N % 2 == 0:
        n = N - 1
        even = True
    else:
        n = N
        even = False

    for i in range(2, n+1):
        if i % 2 == 0:
            print(1, i)
            print(N+i+1, N+i)
        else:
            print(i, i-1)
            print(N+i, 1)
    print(3, N+1)
    if even:
        k //= 2
        print(N, k)
        a = N ^ k ^ 1
        print(a + N, N * 2)
    return


# Generated by 1.1.4 https://github.com/kyuridenamida/atcoder-tools  (tips: You use the default template now. You can remove this line by using your custom template)
def main():
    def iterate_tokens():
        for line in sys.stdin:
            for word in line.split():
                yield word
    tokens = iterate_tokens()
    N = int(next(tokens))  # type: int
    solve(N)

if __name__ == '__main__':
    main()