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>
#define REP(i,n) for(int i=0;i<(int)(n);i++)
#define ALL(x) (x).begin(),(x).end()
#define LL long long

using namespace std;

int main(){

    int N;cin>>N;
    int a[N];REP(i,N)cin>>a[i];

    int maxl = *max_element(a,a+N);
    int cnt[maxl+1];
    fill_n(cnt,maxl+1,0);
    REP(i,N)cnt[a[i]]++;

    for(int i=maxl;i>(maxl-1)/2;i--){
        cnt[i]-=(maxl%2==0&&i==(maxl-1)/2+1 ? 1:2);
        if(cnt[i]<0){
            cout<<"Impossible"<<endl;
            return 0;
        }
    }
    for(int i=(maxl-1)/2+1;i>=0;i--){
        if(cnt[i]){
            cout<<"Impossible"<<endl;
            return 0;
        }
    }
    cout<<"Possible"<<endl;

    return 0;
}