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
readline = sys.stdin.readline       
N = int(readline())

ans = 'Yes' if -N&N != N else 'No'
print(ans)
if ans == 'Yes':
    Edge = []
    
    for i in range(2, N-(N%2 == 0)+1):
        Edge.append((1, i))
        if not i&1:
            Edge.append((i, N+i+1))
            Edge.append((i+1, N+i))
    Edge.append((N+1, N+3))
    if not N&1:
        Edge.append(((-N&N)+1, N))
        Edge.append((N^(-N&N), N+N))
    print('\n'.join([' '.join(map(str, e)) for e in Edge]))