Problem:
Score : 700 points

Problem Statement

Aoki loves numerical sequences and trees.

One day, Takahashi gave him an integer sequence of length N, a_1, a_2, ..., a_N, which made him want to construct a tree.

Aoki wants to construct a tree with N vertices numbered 1 through N, such that for each i = 1,2,...,N, the distance between vertex i and the farthest vertex from it is a_i, assuming that the length of each edge is 1.

Determine whether such a tree exists.

Constraints

2 ≦ N ≦ 100

1 ≦ a_i ≦ N-1

Input

The input is given from Standard Input in the following format:

N
a_1 a_2 ... a_N

Output

If there exists a tree that satisfies the condition, print Possible. Otherwise, print Impossible.

Sample Input 1

5
3 2 2 3 3

Sample Output 1

Possible

The diagram above shows an example of a tree that satisfies the conditions. The red arrows show paths from each vertex to the farthest vertex from it.

Sample Input 2

3
1 1 2

Sample Output 2

Impossible

Sample Input 3

10
1 2 2 2 2 2 2 2 2 2

Sample Output 3

Possible

Sample Input 4

10
1 1 2 2 2 2 2 2 2 2

Sample Output 4

Impossible

Sample Input 5

6
1 1 1 1 1 5

Sample Output 5

Impossible

Sample Input 6

5
4 3 2 3 4

Sample Output 6

Possible

Solution in Ruby (2.3.3) (judge verdict: Accepted)

N = gets.to_i
A = gets.split.map(&:to_i)

counter = Hash.new(0)
max_dist = 0

A.each do |a|
  max_dist = a if max_dist < a
  counter[a] += 1
end

i = 0
j = max_dist

while i <= j
  if i != j
    counter[j] -= 2
  else
    counter[j] -= 1
  end

  if counter[j + 1] > 0
    counter[j + 1] = 0
  end

  i += 1
  j -= 1
end

if counter.values.all? { |x| x == 0 }
  puts 'Possible'
else
  puts 'Impossible'
end