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:
# coding: utf-8
# Your code here!

import sys
sys.setrecursionlimit(10**6)
readline = sys.stdin.readline #文字列入力のときは注意

n = int(input())
#a = [int(i) for i in readline().split()]
m = n
while m%2 ==0:
    m//=2
if m == 1:
    print("No")
    exit()
elif n%2:
    print("Yes")
    for i in range(2,n):
        print(i, i+1)
    print(n,1)
    print(1,n+n-1)
    for i in range(2,n,2):
        print(i+n,i+n+1)
    for i in range(2,n-2,2):
        print(i+n,i+n+3)
    print(n+1,n-1)
else:
    m=n-1
    print("Yes")
    for i in range(2,m):
        print(i, i+1)
    print(m,1)
    print(1,m+n-1)
    for i in range(2,m,2):
        print(i+n,i+n+1)
    for i in range(2,m-2,2):
        print(i+n,i+n+3)
    print(n+1,m-1)
    
    c = n^(n-2)^1
    print(c,n)
    if ((n-2)-c-1)%4==2:
        print(n-2,n+n)    
    else:
        print(n+1,n+n)