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)

def solve
  if A.size==2
    return A[1]==1 ? :Possible : :Impossible
  end
  if A[0]==1
    return :Impossible if A[1]==1
    return A[-1]==2 ? :Possible : :Impossible
  end
  A.each_cons(2) do |a,b|
    return :Impossible unless a==b || a+1==b
  end
  # 中心が1個
  if A[0]!=A[1]
    return :Impossible unless A[0]+A[0]==A[-1]
    h=Hash.new{0}
    A[1..-1].each{|i|h[i]+=1}
    return h.find{|k,v|v==1} ? :Impossible : :Possible
  end
  # 中心が2個
  return :Impossible if A[0]==A[2]
  return :Impossible unless A[0]+A[0]-1==A[-1]
  h=Hash.new{0}
  A[2..-1].each{|i|h[i]+=1}
  h.find{|k,v|v==1} ? :Impossible : :Possible
end

gets
A=gets.split.map(&:to_i).sort
puts solve