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: Wrong Answer)

n = gets.chomp.to_i

dis = gets.chomp.split(" ").map(&:to_i)

dis.sort!

max_height = dis[0]

if dis[1] == 1 and n > 2
  puts "Impossible"
  exit(0)
end

if dis[-1] > max_height * 2
  puts "Impossible"
  exit(0)
end


level = {}

dis.each do |d|
  l = d - max_height
  level[l] = (level[l] || 0) + 1
end

(2..max_height).each do |l|
  if !level[l]  or level[l] ==0 or  (level[l] > level[l - 1])
    puts "Impossible"
    exit(0)
  end
end

puts "Possible"