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())
b = N.bit_length()
if 2 ** (b - 1) == N:
    print('No')
    exit()

ans = [(N + 2, N + 1), (N + 2, 3), (3, 1), (1, 2), (2, 3 + N)]
for i in range(4, N, 2):
    ans.append((1, i))
    ans.append((1, i + 1))
    ans.append((i, N + i + 1))
    ans.append((i + 1, N + i))

if N % 2 == 0:
    for i in range(30, -1, -1):
        if N > 2 ** i:
            a = 2 ** i
            break
    b = N - a + 1
    ans.append((N, a))
    ans.append((b, 2 * N))
    

print('Yes')
for a, b in ans:
    print(a, b)