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++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <stdio.h>

int N,cnt[101];

int main()
{
	scanf ("%d",&N);
	for (int i=0;i<N;i++){
		int x; scanf ("%d",&x);
		cnt[x]++;
	}

	int mx;
	for (mx=N;mx>=0;mx--) if (cnt[mx]) break;

	if (cnt[mx] == 1){
		puts("Impossible");
		return 0;
	}

	for (int i=(mx-1)/2;i>=1;i--) if (cnt[i]){
		puts("Impossible");
		return 0;
	}

	if (mx % 2 == 0 && cnt[mx/2] != 1){
		puts("Impossible");
		return 0;
	}

	for (int i=mx/2+1;i<=mx;i++) if (cnt[i] == 1){
		puts("Impossible");
		return 0;
	}

	puts("Possible");
	return 0;
}