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 Java (OpenJDK 11.0.6) (judge verdict: Wrong Answer)

import java.util.Scanner;

public class Main {
	public static void main(String[] args) {
		Scanner sc = new Scanner(System.in);
		int n = sc.nextInt();
		int a[] = new int[n];
		int max = 0;
		int freq[] = new int[n];
		for(int i=0;i<n;i++) {
			a[i] = sc.nextInt();
			max = Math.max(max,a[i]);
			freq[a[i]]++;
		}
		if(max==1) {
			if(n>2) {
				System.out.println("Impossible");
			}
			else {
				System.out.println("Possible");
			}
			return;
		}
		if(max%2==0) {
			if(freq[max/2]!=1) {
				System.out.println("Impossible");
				return;
			}
			for(int i=0;i<max/2;i++) {
				if(freq[i]>0) {
					System.out.println("Impossible");
					return;
				}
			}
			for(int i=max/2+1;i<=max;i++) {
				if(freq[i]<2) {
					System.out.println("Impossible");
					return;
				}
			}
			System.out.println("Possible");
			return;
		}
		else {
			for(int i=0;i<(max+1)/2;i++) {
				if(freq[i]>0) {
					System.out.println("Impossible");
					return;
				}
			}
			for(int i=(max+1)/2;i<=max;i++) {
				if(freq[i]<2) {
					System.out.println("Impossible");
					return;
				}
			}
			System.out.println("Possible");
			return;
		}
	}
}