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 PYTHON):

Solution:
N = input()

if N<=2:
  print "No"
  exit(0)

for i in range(100):
  if 2**i==N:
    print "No"
    exit(0)

print "Yes"

print 1, 3
print 3, 2
print 2, N+1
print N+1, N+3
print N+3, N+2

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

if N%2==0:
  print N, N-2
  x = N^(N-2)^1
  if x==3:
    print 3, N+N
  else:
    print x+N, N+N