Problem:
Score : 700 points

Problem Statement

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.

Sample Input 1

3

Sample Output 1

Yes
1 2
2 3
3 4
4 5
5 6

The sample output represents the following graph:

Sample Input 2

1

Sample Output 2

No

There is no tree satisfying the condition.

Solution in Ruby (2.3.3) (judge verdict: Wrong Answer)

def pp(a,b)
  puts [a,b].join(" ")
end

n = gets.to_i
k = Math.log2(n)
if n == 1 || k == k.to_i
  puts "No"
  exit
end

puts "Yes"
pp(1,2)
pp(2,3)
pp(3,n+1)
pp(n+1,n+2)
pp(n+2,n+3)

((n-3)/2).times do |i|
  t = i*2
  pp(t+4,t+5)
  pp(t+5,1)
  pp(1,n+t+4)
  pp(n+t+4,n+t+5)
end

if n % 2 == 0
  t = 2**(k.to_i)
  pp(n,n+t)
  pp(n-t+1,2*n)
end