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<=2 or n==4:
  print("No")
elif n==3:
  print("Yes")
  for i in range(5):
    print(i+1,i+2)
elif n%2==0:
  nn=n.bit_length()
  if n==2**(nn-1):
    print("No")
  else:
    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):
      print(n+1,i)
      if i%2==0:
        print(i,i+n+1)
      else:
        print(i,i+n-1)
    n1,n2=2**(nn-1),(n+1)-2**(nn-1)
    print(n1,n)
    print(n2,2*n)
else:
  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+1):
    print(1,i)
    if i%2==0:
      print(i,i+n+1)
    else:
      print(i,i+n-1)