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 lint=long long;
void err(){
    std::cout<<"Impossible"<<'\n';
    exit(0);
}
int main(){
    std::cin.tie(nullptr);std::ios_base::sync_with_stdio(false);
    std::cout.setf(std::ios_base::fixed);std::cout.precision(15);
    lint n;std::cin>>n;
    std::vector<lint>a(n);
    for(lint i=0;i<n;i++){
        lint x;std::cin>>x;
        if(n<=x)err();
        a.at(x)++;
    }
    lint max=0;
    for(lint i=0;i<n;i++){
        if(a.at(i))max=i;
    }
    if(max==1&&n!=2)err();
    lint min=(max+1)/2;
    for(lint i=1;i<min;i++){
        if(a.at(i))err();
    }
    if(a.at(min)!=max%2+1)err();
    for(lint i=min+1;i<=max;i++){
        if(a.at(i)<2)err();
    }
    std::cout<<"Possible"<<'\n';
}