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

N = int(input())

if (N & -N) == N:
    print('No')
    sys.exit()

N2 = N if N%2 else N-1
anss = []
for i in range(2, N2+1):
    anss.append((1, i))
for i in range(2, N2+1):
    anss.append((i, (i^1)+N))
anss.append((2+N, 1+N))

if N%2 == 0:
    N2 = N+1
    k = 1<<(N2.bit_length()-1)
    anss.append((N, k))
    anss.append((2*N, N2^k))

print('Yes')
print('\n'.join(map(lambda x: str(x[0])+' '+str(x[1]), anss)))