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-1)==0:
    print("No")
else:
    print("Yes")
    if N%2==1:
        k=N//2
        for i in range(1,k+1):
            print(2*i,2*i+1)
            print(2*i+1,1)
            print(1,2*i+N)
            print(2*i+N,2*i+1+N)
        print(1+N,2)
    else:
        k=N//2-1
        for i in range(1,k+1):
            print(2*i,2*i+1)
            print(2*i+1,1)
            print(1,2*i+N)
            print(2*i+N,2*i+1+N)
        print(1+N,2)
        n=N.bit_length()-1
        a=2**n
        b=N+1-a
        print(N,a+N)
        print(2*N,b)