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())

last = -1
tmp = N
ct = 0
while tmp > 0:
    if last == -1:
        if tmp&1:
            last = ct
    tmp = tmp>>1
    ct += 1

if 1 << last == N:
    print("No")
elif last == 0:
    print("Yes")
    for i in range(2, N+1):
        if i&1:
            print(1, i)
            print(i, N+i-1)
        else:
            print(1, i)
            print(i, N+i+1)
    print(N+3, N+1)
else:
    print("Yes")
    for i in range(2, N):
        if i&1:
            print(1, i)
            print(i, N+i-1)
        else:
            print(1, i)
            print(i, N+i+1)
    print(N+3, N+1)
    k = 1 << last
    l = N ^ (k+1)
    print(k, N)
    print(l, 2*N)