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())

res = 'Yes'

k = 0

while 2 ** (k + 1) - 1 <= N:
    k += 1

K = 2 ** k - 1


if N == 1:
    res = 'No'
if N == K + 1:
    res = 'No'
# print(N, K)

if N % 2 == 0:
    M = N - 1
else:
    M = N
if (M - K) % 4 == 0 and M > K:
    L = M - 2
else:
    L = M

P = N - K
if P % 2 == 0:
    P = 2

print(res)

if res == 'Yes':
    line = list(range(1, K + 1))
    line.remove(P)
    line = [1, P] + line[1:]
    
    for i in range(K - 1):
        print(line[i], line[i + 1])
    print(line[-1], N + 1)
    for i in range(K - 1):
        print(N + line[i], N + line[i + 1])
    if L > K:
        print(1, K + 1)
        for i in range(K + 1, L):
            print(i, i + 1)
        for i in range(K + 1, L):
            print(N + i, N + i + 1)
        print(N + L, 1)
    if M > L:
        print(N + 1, M - 1)
        print(M - 1, M)
        print(N + M - 1, N + M)
        print(N + M, N + 1)
    if N > M:
        print(N, K + 1)
        print(N + N, P)