Problem:
Score : 800 points

Problem Statement

In Finite Encyclopedia of Integer Sequences (FEIS), all integer sequences of lengths between 1 and N (inclusive) consisting of integers between 1 and K (inclusive) are listed.

Let the total number of sequences listed in FEIS be X. Among those sequences, find the (X/2)-th (rounded up to the nearest integer) lexicographically smallest one.

Constraints

1 \leq N,K \leq 3 × 10^5

N and K are integers.

Input

Input is given from Standard Input in the following format:

K N

Output

Print the (X/2)-th (rounded up to the nearest integer) lexicographically smallest sequence listed in FEIS, with spaces in between, where X is the total number of sequences listed in FEIS.

Sample Input 1

3 2

Sample Output 1

2 1

There are 12 sequences listed in FEIS: (1),(1,1),(1,2),(1,3),(2),(2,1),(2,2),(2,3),(3),(3,1),(3,2),(3,3).
The (12/2 = 6)-th lexicographically smallest one among them is (2,1).

Sample Input 2

2 4

Sample Output 2

1 2 2 2

Sample Input 3

5 14

Sample Output 3

3 3 3 3 3 3 3 3 3 3 3 3 2 2

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

def o()
  m = (K/2) + 1
  a = [m] * N
  d = N - 1
  until d<=0
    d -= 2
    a[-1] -= 1
    next if a[-1] != 0
    (-1).downto(-N) do |i|
      a[i] = K
      d -= 1 if a[i-1] == m
      a[i-1] -= 1
      break if a[i-1] != 0
    end
  end
  a*" "
end

def e()
  m = K/2
  m.to_s + " #{m+1}"*(N-1)
end

K, N = gets.split.map &:to_i
if N<=2
  puts (K+1)/2
elsif K==1
  puts ([K]*(N/2))*" "
else
  puts K.odd? ? o : e
end