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: Wrong Answer)

using System;
using System.Collections.Generic;
using System.Collections;
using System.Linq;


namespace tes
{
	class contest
	{
		static void Main(string[] args)
		{
			 
            var num = int.Parse(Console.ReadLine());
            var input = Console.ReadLine().Split(' ').Select(int.Parse).ToArray();

            int max = input.Max();
            int min = input.Min();
            int midCount = input.Length - input.Count(c => c == max);
            Array.Sort(input);
            Array.Reverse(input);
            var tmp = new int[max+1][];
            tmp[0] = new int[] { max };
            tmp[tmp.Length - 1] = new int[] { max };

            bool check = true;
            if (input.Count(c => c == max) < 2 || (num >= 3 && input.Count(c => c == 1) >= 2)) check = false;

           
            
            for (int i = 0; i < input.Length; i++)
            {
                int current = input[i];
                if (current == max) continue;
				if (current < (max + 1) / 2)
                {
                    check = false;
                    break;
                }
				
                if(tmp[current]==null)tmp[current] = new int[] {current };
                else if(tmp[max - current]== null) tmp[max-current] = new int[] { current};
                else
                {
                    for (int j = 0; j < tmp.Length; j++)
                    {
                        if (j+1 == current) 
                        {
                            break;
                        }

                        if (j == tmp.Length - 1) check = false;
                    }
                }
            }
            

            for(int i =0; i<tmp.Length; i++)
            {
                if(tmp[i]==null)
                {
                    check = false;
                    break;
                }
            }

            if (!check ) Console.WriteLine("Impossible");
            else Console.WriteLine("Possible");

		}				 
	}
}