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 main():
    import sys
    input = sys.stdin.readline

    N = int(input())
    k = N.bit_length()

    if 1<<(k-1) == N:
        print('No')
        exit()
    print('Yes')
    if N == 3:
        for i in range(1, 6):
            print(i, i+1)
        exit()

    N0 = 1<<(k-1)
    e = [(N0-1, N+1)]
    for i in range(1, N0-1):
        e.append((i, i+1))
        e.append((i+N, i+1+N))

    if N%2 == 0:
        v = N ^ N0
        e.append((N, v))
        e.append((N*2, N0))
        for i in range(N0, N):
            if i % 2 == 0:
                e.append((i, v))
                e.append((i + N, i+N+1))
            else:
                e.append((i, i - 1))
                e.append((i + N, v + 1))
    else:
        for i in range(N0, N + 1):
            if i % 2 == 0:
                e.append((i, 2))
                e.append((i + N, i + 1 + N))
            else:
                e.append((i, i - 1))
                e.append((i + N, 3 + N))
    for edge in e:
        print(*edge)


if __name__ == '__main__':
    main()