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())
pow2s = [2**item for item in range(32)]
if n in pow2s:
    print("No")
    exit()
print("Yes")
print(n+1, n+2)
for i in range(3, n+1, 2):
    print(1, i)
    print(1, i - 1)
    print(i, n + i - 1)
    print(i - 1, n + i)
if n % 2 == 0:
    upper = 1 << (n.bit_length() - 1)
    lower = n ^ 1 ^ upper
    print(upper, n)
    print(lower, n + n)