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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace c
{
    class Program
    {
        static void Main(string[] args)
        {
            int n = int.Parse(Console.ReadLine());
            int[] a = Console.ReadLine().Split().Select(int.Parse).ToArray();
            int max = 0;
            for (int i = 0; i < n; i++)
            {
                max = Math.Max(max, a[i]);
            }
            int[] ls = new int[max + 1];
            for (int i = 0; i < n; i++)
            {
                ls[a[i]]++;
            }
            if(max % 2 ==0)
            {
                for (int i = 0; i < max/2; i++)
                {
                    if(ls[i]>0)
                    {
                        Console.WriteLine("Impossible");
                        return;
                    }
                }
                if (ls[max / 2] != 1)
                {
                    Console.WriteLine("Impossible");
                    return;
                }
                for (int i = max/2 + 1; i <= max; i++)
                {
                    if(ls[i]<2)
                    {
                        Console.WriteLine("Impossible");
                        return;
                    }
                }
            }
            else
            {
                for (int i = 0; i < (max + 1)/ 2; i++)
                {
                    if (ls[i] > 0)
                    {
                        Console.WriteLine("Impossible");
                        return;
                    }
                }
                if (ls[(max + 1) / 2] != 2)
                {
                    Console.WriteLine("Impossible");
                    return;
                }
                for (int i = (max + 1) / 2 + 1; i <= max; i++)
                {
                    if (ls[i] < 2)
                    {
                        Console.WriteLine("Impossible");
                        return;
                    }
                }
            }
            Console.WriteLine("Possible");
        }
    }
}