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())
if n < 3 or format(n - 1, "b").count("0") == 0:
    print("No")
    exit()
else:
    print("Yes")
for i in range(1, 3):
    print(i, i + 1)
    print(i + n, i + 1 + n)
print(1, 3 + n)

for i in range(5, n + 1, 2):
    print(1, i - 1)
    print(i - 1, i)
    print(1, i + n)
    print(i + n, i - 1 + n)

if not n % 2:
    a = 1 << (len(format(n + 1, "b")) - 1)
    b = (n + 1) ^ a
    a = a + n if a % 2 else a
    b = b + n if b % 2 else b
    print(a, n)
    print(b, 2 * n)