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:
N = int(input())

M = N
while M % 2 == 0 :
    M //= 2
    
if M == 1 or N < 3 :
    print('No')
else :
    print('Yes')
    print(3, 1)
    print(1, 2)
    print(2, 3 + N)
    print(3 + N, 1 + N)
    print(1 + N, 2 + N)
        
    for i in range(4, N, 2) :
        print(1, i + 1)
        print(i + 1, i)
        print(1, N + i)
        print(N + i, N + i + 1)
        
    if N % 2 == 0 :
        i = 1
        while True :
            if N >> i & 1 :
                break
            i += 1
        print((1 << i) + 1, N)
        print(2 * N - (1 << i), 2 * N)