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:
import sys

N=int(input())
n=N
while n%2==0:
    n//=2
if n==1:
    print("No")
    sys.exit()

m=0
while True:
    m+=1
    if 2**m>N:
        m-=1
        break

ans=[]
if N%2==0:
    ans.append([N,2**m])
    ans.append([2*N,(N-2**m+1+N)])

ans.append([N+1,3])
for i in range(1,(N-1)//2+1):
    num=2*i
    ans.append([1,num])
    ans.append([num,num+1])
    ans.append([1,N+num+1])
    ans.append([N+num+1,N+num])
    
print("Yes")
for u in ans:
    print(*u)