Problem:
Score : 400 points

Problem Statement

Snuke has an empty sequence a.

He will perform N operations on this sequence.

In the i-th operation, he chooses an integer j satisfying 1 \leq j \leq i, and insert j at position j in a (the beginning is position 1).

You are given a sequence b of length N. Determine if it is possible that a is equal to b after N operations. If it is, show one possible sequence of operations that achieves it.

Constraints

All values in input are integers.

1 \leq N \leq 100

1 \leq b_i \leq N

Input

Input is given from Standard Input in the following format:

N
b_1 \dots b_N

Output

If there is no sequence of N operations after which a would be equal to b, print -1.
If there is, print N lines. In the i-th line, the integer chosen in the i-th operation should be printed. If there are multiple solutions, any of them is accepted.

Sample Input 1

3
1 2 1

Sample Output 1

1
1
2

In this sequence of operations, the sequence a changes as follows:

After the first operation: (1)

After the second operation: (1,1)

After the third operation: (1,2,1)

Sample Input 2

2
2 2

Sample Output 2

-1

2 cannot be inserted at the beginning of the sequence, so this is impossible.

Sample Input 3

9
1 1 1 2 2 1 2 3 2

Sample Output 3

1
2
2
3
1
2
2
1
1

Solution in Ruby (2.3.3) (judge verdict: Time Limit Exceeded)

n = gets.chomp.to_i
as = gets.split.map(&:to_i)

ns = as.reduce({}) { |h, a|
  if h[a]
    h[a] += 1
  else
    h[a] = 1
  end
  h
}

memo = {}

slv = ->i,xs {
  return memo[xs] if memo[xs]
  if i == n+1
    ys = []
    xs.each do |x| ys.insert(x-1, x) end
    memo[xs] =
      if ys == as
        xs
      else
        nil
      end
  else
    (1..i).each do |j|
      next if ns[j].nil? || ns[j].zero?
      ns[j] -= 1
      if r = slv.(i+1, xs.dup.insert(j-1, j))
        return memo[xs] = r
      end
      ns[j] += 1
    end
    memo[xs] = nil
  end
}

if r = slv.(1, [])
  puts r
else
  puts "-1"
end