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)
cnt = Hash.new(0)
n.times do |i|
    cnt[a[i]] += 1
end
b = cnt.keys.sort
if !((b[-1].even? && b[0] == b[-1]/2 && cnt[b[0]] == 1) || (b[-1].odd? && b[0] == (b[-1]+1)/2 && cnt[b[0]] == 2)) || b.size-1 != b[-1]-b[0] || (b[-1] == 1 && n > 2)
    puts "Impossible"
    exit
end
b[1..-1].each do |num|
    if cnt[num] < 2
        puts "Impossible"
        exit
    end
end
puts "Possible"