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();

            var dp = new int[num, num];
            var memo = new int[num];

            bool p = true;
            if (num >= 3 && input[0] == 1 && input[1] == 1)
            {
                p = false;
            }
            else
            {
                
                memo[0] = input[0];
                for (int i = 1; i < num; i++)
                {
                    if(input[i]>= i)
                    {
                        memo[i] = Math.Max(input[i], i);
                        continue;
                    }
                    else
                    {
                        if(input[i-1] == input[i])
                        {
                            memo[i] = memo[i-1];
                        }
                        else
                        {
                            p = false;
                            break;
                        }
                    }

                }
            }

            if (memo[0]+1 < memo[memo.Length - 1]) p = false;          


            if (p) Console.WriteLine("Possible");
            else Console.WriteLine("Impossible");
		}				 
	}
}