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# (.NET Core 3.1.201) (judge verdict: Wrong Answer)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Text.RegularExpressions;
using static System.Console;
using static System.Math;
using static MyIO;
using static MyUtil;

public class C { public static void Main() => (new Solver()).Solve(); }

public class Solver
{
	public void Solve()
	{
		long N = GetLong();
		var a = GetLongs(N);

		long D = a.Max();
		var cnt = new long[D + 1];
		for(long i = 0; i < N; i++)
			cnt[a[i]]++;

		for(long i = 0; i <= D; i++)
		{
			long d = Max(i, D - i);
			if(cnt[d] == 0)
			{
				WriteLine("Impossible");
				return;
			}
			cnt[d]--;
		}
		for(long i = 0; i < D / 2 + 1; i++)
		{
			if(cnt[i] > 0)
			{
				WriteLine("Impossible");
				return;
			}
		}
		WriteLine("Possible");
	}
}

public static class MyIO
{
	private static string[] args = new string[0];
	private static int cnt = 0;
	private static IEnumerable<string> getArgs(long N) => Range(0, N).Select(_ => getArg());
	private static string getArg()
	{
		if(cnt == args.Length) { args = Console.ReadLine().Split(); cnt = 0; }
		return args[cnt++];
	}

	public static int GetInt() => int.Parse(getArg());
	public static long GetLong() => long.Parse(getArg());
	public static double GetDouble() => double.Parse(getArg());
	public static decimal GetDecimal() => decimal.Parse(getArg());
	public static string GetString() => getArg();
	public static char GetChar() => getArg()[0];
	public static int[] GetInts(long N) => getArgs(N).Select(int.Parse).ToArray();
	public static long[] GetLongs(long N) => getArgs(N).Select(long.Parse).ToArray();
	public static double[] GetDoubles(long N) => getArgs(N).Select(double.Parse).ToArray();
	public static decimal[] GetDecimals(long N) => getArgs(N).Select(decimal.Parse).ToArray();
	public static string[] GetStrings(long N) => getArgs(N).ToArray();
}

public static class MyUtil
{
	public static void chmax<T>(ref T x, T y) where T : IComparable<T> { if(x.CompareTo(y) < 0) x = y; }
	public static void chmin<T>(ref T x, T y) where T : IComparable<T> { if(x.CompareTo(y) > 0)	x = y; }
	public static void swap<T>(ref T x, ref T y) { T tmp = x; x = y; y = tmp; }
	public static long rightShift(long x, long y) => (x >> (int)y);
	public static long leftShift(long x, long y) => (x << (int)y);
	public static bool contains(long s, long x) => (rightShift(s, x) & 1L) == 1L;
	public static long gcd(long x, long y) => (y == 0) ? x : gcd(y, x % y);
	public static long lcm(long x, long y) => (x / gcd(x,y)) * y;	
	public static IEnumerable<int> Range(long start, long count) => Enumerable.Range((int)start, (int)count);
	public static void Sort(this List<string> source) => source.Sort(StringComparer.Ordinal);
}