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: Accepted)

#include <cstdio>
#include <iostream>
inline int Get() {
	char ch;
	while ((ch = getchar()) < '0' || ch > '9');
	int Num = ch - '0';
	while ((ch = getchar()) >= '0' && ch <= '9')
		Num = (Num << 3) + (Num << 1) + ch - '0';
	return Num;
}
const int N = 105;
int n, w[N], mn, mx;
int main() {
	n = Get(), mn = n, mx = 0;
	for (int i = 1, x; i <= n; ++i) 
		++w[x = Get()], mn = std :: min(mn, x), mx = std :: max(mx, x);
	
	if (w[mx] < 2) {
		puts("Impossible");
		return 0;
	}
	
	if (mx & 1) 
		if (mn != (mx + 1) / 2 || w[mn] != 2) {
			puts("Impossible");
			return 0;
		}
	
	if (!(mx & 1))
		if (mn != mx / 2 || w[mn] != 1) {
			puts("Impossible");
			return 0;
		}
	
	for (int i = mn + 1; i < mx; ++i) 
		if (w[i] < 2) {
			puts("Impossible");
			return 0;
		}
	
	puts("Possible");
	return 0;
}