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 Java8 (OpenJDK 1.8.0) (judge verdict: Wrong Answer)

import java.util.Scanner;

public class Main {

	static Scanner sc = new Scanner(System.in);

	public static void main(String[] args) {
		System.out.println(solve() ? "Possible" : "Impossible");
	}

	static boolean solve() {
		int N = sc.nextInt();
		int[] A = new int[N];
		int[] hist = new int[N];
		int max = 0;
		for (int i = 0; i < N; ++i) {
			A[i] = sc.nextInt();
			hist[A[i]]++;
			max = Math.max(max, A[i]);
		}
		if (hist[max] == 1) return false;
		if (max == 1) {
			return N == 2;
		}
		if (max == 2) {
			return hist[1] == 1;
		}
		for (int i = max; i > max / 2; --i) {
			if (hist[i] <= 1) return false;
		}
		if (max % 2 == 0) {
			if (hist[max / 2] == 0) return false;
		}
		for (int i = 1; i <= (max - 1) / 2; ++i) {
			if (hist[i] != 0) return false;
		}
		return true;
	}

}