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())

if N & (N-1) == 0:
  print("No")
  exit()

print("Yes")
print(1, 2)
print(2, 3)
print(3, N+1)
print(N+1, N+2)
print(N+2, N+3)

for i in range(4, N, 2):
  print(i, i+1)
  print(i+1, N+1)
  print(N+1, N+i)
  print(N+i, N+i+1)

if N % 2 == 0:
  print(N, N-1)
  x = N^(N-1)^1
  if x % 2 == 0:
    print(N+x, 2*N)
  else:
  	print(x, 2*N)