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:
def is_2power(n: int)->bool:
    a = 1
    while a < n:
        a <<= 1
    return a == n


def skolem_xor_tree(N: int)->list:
    if is_2power(N):
        return []

    res = [
        (1, 2),
        (2, 3),
        (3, N+1),
        (N+1, N+2),
        (N+2, N+3),
    ]

    for a in range(4, N+(N % 2), 2):
        res.append((1, a))
        res.append((a, a+1))

        res.append((1, N+a+1))
        res.append((N+a+1, N+a))

    if N % 2 == 0:
        a = N ^ 1
        b = None
        for i in range(2, N):
            if (a ^ i) < N:
                a = a ^ i
                b = i
                break

        if not b:
            raise Exception()

        if a < N and a % 2 == 1:
            a += N

        if b < N and b % 2 == 1:
            b += N

        res.append((a, N))
        res.append((b, N+N))

    return res


if __name__ == "__main__":
    N = int(input())

    ans = skolem_xor_tree(N)
    if ans:
        print('Yes')
        for a, b in ans:
            print(a, b)
    else:
        print('No')