Problem:
Score : 500 points

Problem Statement

You are given an integer N. Determine if there exists a tuple of subsets of \{1,2,...N\}, (S_1,S_2,...,S_k), that satisfies the following conditions:

Each of the integers 1,2,...,N is contained in exactly two of the sets S_1,S_2,...,S_k.

Any two of the sets S_1,S_2,...,S_k have exactly one element in common.

If such a tuple exists, construct one such tuple.

Constraints

1 \leq N \leq 10^5

N is an integer.

Input

Input is given from Standard Input in the following format:

N

Output

If a tuple of subsets of \{1,2,...N\} that satisfies the conditions does not exist, print No.
If such a tuple exists, print Yes first, then print such subsets in the following format:

k
|S_1| S_{1,1} S_{1,2} ... S_{1,|S_1|}
:
|S_k| S_{k,1} S_{k,2} ... S_{k,|S_k|}

where S_i={S_{i,1},S_{i,2},...,S_{i,|S_i|}}.

If there are multiple such tuples, any of them will be accepted.

Sample Input 1

3

Sample Output 1

Yes
3
2 1 2
2 3 1
2 2 3

It can be seen that (S_1,S_2,S_3)=(\{1,2\},\{3,1\},\{2,3\}) satisfies the conditions.

Sample Input 2

4

Sample Output 2

No

Solution in Ruby (2.7.1) (judge verdict: Wrong Answer)

n=gets.to_i
k=1
i=1
s=[]
while k<=100000
  if n==k
    puts "Yes"
    puts k
    (i+1).times do
      s << []
    end
    c=0
    cc=i-1
    (i/3).times do |a|
      3.times do |j|
        (i-3*a).times do |b|
          s[3*a+j] << a+cc*j+b+1+c
        end
      end
      s[3*a+2][-1]=s[3*a][3*a]
      (i+1-3*(a+1)).times do |j|
        3.times do |b|
          s[3*(a+1)+j] << s[3*a+b][3*a+j+1]
        end
      end
      c=c+cc*3-1
      cc-=3
    end
    if i%3==1
      s[-2] << c+(i/3)+1
      s[-1] << c+(i/3)+1
    elsif i%3==2
      s[-3] << c+(i/3)+1
      s[-3] << c+(i/3)+2
      s[-2] << c+(i/3)+2
      s[-2] << c+(i/3)+3
      s[-1] << c+(i/3)+3
      s[-1] << c+(i/3)+1
    end
    (i+1).times do |j|
      puts [i,s[j].join(" ")].join(" ")
    end
    exit
  end
  i+=1
  k+=i
end
puts "No"