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
import math

input = sys.stdin.readline
sys.setrecursionlimit(100000)

def getN():
    return int(input())

def getlist():
    return list(map(int, input().split()))

n = getN() #22

dig_num = math.floor(math.log2(n + 1))  # 5
# print(dig_num)
sen_end = 2 ** (dig_num) - 1  # 15

if 2 ** (dig_num) == n or n == 1:
    print("No")
    sys.exit(0)
else:
    print("Yes")

for i in range(sen_end - 1):
    print(i+1, i+2)  # 1-2, 2-3, ..., 14-15
    print(i+1+n, i+2+n)  # 23-24, 24-25, ..., 36-37

print(n + 1, sen_end)

uni_end = n
if n % 2 == 0:
    uni_end -= 1  # 21

for i in range((uni_end - sen_end) // 2):  # 3 times
    print(1, sen_end + i * 2 + 1)  # 1-16
    print(1, sen_end + i * 2 + 2 + n)  # 1-39
    print(sen_end + i * 2 + 1 + n, sen_end + i * 2 + 2 + n)  # 38-39
    print(sen_end + i * 2 + 1, sen_end + i * 2 + 2)  # 16-17

if n % 2 == 0:
    one_count = 0
    for idx, digit in enumerate(bin(n)[2:]):
        # print(digit, one_count)
        if one_count == 0:
            # print(digit)
            if digit == "1":
                one_count += 1
        else:
            if digit == "1":
                one_goal = dig_num - idx  # second "1" from upper digit
                break

    if one_goal != 1:
        print(n, 2**one_goal)
        print(2*n, n - 2**one_goal)

    else:
        print(n, 2 ** one_goal)
        print(2 * n, 2 * n - 2 ** one_goal + 1)