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:
import sys
n = int(input())
if n < 3:
	print("No")
else:
	#print("Yes")
	ans = []
	for i in range(2, n, 2):
		ans.append([1, i])
		ans.append([i, i+1])
		ans.append([1, n+i+1])
		ans.append([n+i+1, n+i])
	ans.append([3, n+1])
	if n%2 == 0:
		for i in range(1, n):
			if (n+1)^i < n:
				if i%2 == 0:
					ans.append([i, n])
					ans.append([((n+1)^i) + n, 2*n])
				else:
					ans.append([(n+1)^i, n])
					ans.append([n+i, 2*n])
				break
		else:
			print("No")
			sys.exit()
	print("Yes")
	for a in ans:
		print(*a)