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 <bits/stdc++.h>
using namespace std;
int N, M;
map<int,int> a;
bool g(int L) {
	vector<int> c(L);
	int mi = 1<<30, ma = -1;
	for (int i = 0; i < L; ++ i) {
		c[i] = max(i, L-1-i);
		mi = min(mi, c[i]);
		ma = max(ma, c[i]);
	}
	auto b = a;
	for (int x : c) {
		if (b[x] == 0) return false;
		-- b[x];
	}
	for (auto p : b) {
		if (p.second == 0) continue;
		if (!(mi < p.first && p.first <= ma)) return false;
	}
	return true;
}
bool f() {
	for (int l = M+1; l <= N; ++ l) {
		if (g(l)) return true;
	}
	return false;
}
int main() {
	cin >> N;
	for (int i = 0; i < N; ++ i) {
		int x;
		cin >> x;
		++ a[x];
		M = max(M, x);
	}
	cout << (f() ? "Possible" : "Impossible") << endl;
}