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 <bits/stdc++.h>

using namespace std;

#define rep(i,n) for(int (i)=0;(i)<(int)(n);(i)++)

int main(void){
    int N; cin >> N;
    int a[N+10];
    rep(i,N)cin >> a[i];
    sort(a,a + N);
    int separate = (N - 1) - a[N - 1];
    int depth = N - separate;
    map<int,int> mp;
    rep(i,N)mp[a[i]]++;
    if(depth % 2 == 1){
        if(mp[depth / 2] != 1){
            cout << "Impossible" << "\n";
            return 0;
        }
        mp[depth / 2] = 0;
    }
    int cou = 0;
    rep(i,N){
        if(mp[a[i]] == 1){
            cout << "Impossible" << "\n";
            return 0;
        }
        cou += max(mp[a[i]] - 2, 0);
        mp[a[i]] = 0;
    }
    rep(i,N - 1){
        if(abs(a[i] - a[i + 1]) > 1){
            cout << "Impossible" << "\n";
            return 0;
        }
    }
    if(a[0] != depth / 2)cout << "Impossible" << "\n";
    else if(cou == separate)cout << "Possible" << "\n";
    else cout << "Impossible" << "\n";
    return 0;
}