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.IO;
using System.Linq;
using System.Numerics;
using System.Collections;
using System.Collections.Generic;
using System.Text;
using System.Text.RegularExpressions;
using System.Threading.Tasks;
using static System.Math;
using LayoutKind = System.Runtime.InteropServices.LayoutKind;
using Debug = System.Diagnostics.Debug;
using StructLayoutAttribute = System.Runtime.InteropServices.StructLayoutAttribute;
using FieldOffsetAttribute = System.Runtime.InteropServices.FieldOffsetAttribute;
using MethodImplOptions = System.Runtime.CompilerServices.MethodImplOptions;
using MethodImplAttribute = System.Runtime.CompilerServices.MethodImplAttribute;
using System.Runtime.CompilerServices;

static class P
{
    static void Main()
    {
        int n = int.Parse(Console.ReadLine());
        var a = Console.ReadLine().Split().Select(int.Parse).OrderBy(x => x).ToArray();
        int[] counts = new int[n];

        for (int i = 0; i < a.Length; i++) counts[a[i]]++;

        var diameter = a.Max();
        if(diameter % 2 == 0)
        {
            if (counts[diameter / 2] != 1) goto impossible;
            counts[diameter / 2] = 0;
        }
        for (int i = (diameter + 2) / 2; i <= diameter; i++)
        {
            if (counts[i] < 2) goto impossible;
            counts[i] = 0;
        }
        if (counts.Sum() > 0) goto impossible;
        Console.WriteLine("Possible");
        return;
        impossible:Console.WriteLine("Impossible");
    }
}