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)==N:
  print('No')
  exit()
print('Yes')
P=[[] for i in range(2*N)]
X=1
while 2*X-1<=N:
  X*=2
X-=1
for i in range(X-1):
  P[i].append(i+1)
  P[N+i].append(N+i+1)
P[X-1].append(N)
while X+2<=N:
  P[X].append(X+1)
  P[0].append(X+1)
  P[0].append(X+N)
  P[X+N].append(X+N+1)
  X+=2
if X!=N:
  X=N&(-N)
  P[X-1].append(N-1)
  if X==2:
    P[N-X].append(2*N-1)
  else:
    P[2*N-X-1].append(2*N-1)
for i in range(len(P)):
  for j in range(len(P[i])):
    print(i+1,P[i][j]+1)