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())
for i in range(100):
  if n == 1<<i:
    print("No")
    exit()
print("Yes")
l = (n+1).bit_length()-1
ans = []
for i in range(2,2**l):
  ans.append((i-1,i))
  ans.append((n+i-1,n+i))
ans.append((2**l-1,n+1))
m = n-1+(n%2)
for i in range(2**l+1,m+1):
  ans.append((i-1,i))
  ans.append((n+i-1,n+i))
if n != 2**l-1:
  if (m+1)%4 == 0:
    ans.append((1,2**l))
    ans.append((m,n+2**l))
  else:
    ans.append((m+n,1))
    ans.append((1,2**l))
if n>m and (m+1)%4 == 0:
  ans.append((n-2**l,n))
  ans.append((2**l,2*n))
elif n>m:
  ans.append((2,n))
  ans.append((2*n,2*n-1))
for edge in ans:
  print(*edge)