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:
#!/usr/bin/env python3
n = int(input())
t = n
while t > 0 and t % 2 == 0:
    t //= 2
ans = (t != 1)
print(['No', 'Yes'][ans])
if not ans:
    exit()
print(1, 2)
for i in range(2, n, 2):
    print(i, i + 1)
    print(i + 1, n + 1)
    print(n + 1, n + i)
    print(n + i, n + i + 1)
if n % 2 == 0:
    nob = len(bin(n)[2:])
    a = 1 << (nob - 1)
    b = (n + 1) ^ a
    a = n + a if a % 2 == 0 else a
    b = n + b if b % 2 == 0 else b
    print(n, a)
    print(b, 2 * n)