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++ (GCC 9.2.1) (judge verdict: Wrong Answer)

#pragma GCC optimize("O3")
#include<bits/stdc++.h> 
using namespace std;
using ll=long long;
using P=pair<ll,ll>;
template<class T> using V=vector<T>; 
#define fi first
#define se second
#define all(v) (v).begin(),(v).end()
const ll inf=(1e18);
//const ll mod=998244353;
const ll mod=1000000007;
ll GCD(ll a,ll b) {return b ? GCD(b,a%b):a;}
ll LCM(ll c,ll d){return c/GCD(c,d)*d;}
struct __INIT{__INIT(){cin.tie(0);ios::sync_with_stdio(false);cout<<fixed<<setprecision(15);}} __init;
template<class T> bool chmax(T &a, const T &b) { if (a<b) { a=b; return 1; } return 0; }
template<class T> bool chmin(T &a, const T &b) { if (a>b) { a=b; return 1; } return 0; }
int main(){
    int n;
    cin>>n;
    V<int> a(n);
    for(int i=0;i<n;i++)cin>>a[i];
    sort(all(a));
    ll d=*max_element(all(a));
    if(d%2==0){
        int cnt=0;
        for(int i=0;i<n;i++){
            if(a[i]<=d/2){
                cnt++;
                if(cnt>1||a[i]<d/2){
                    cout<<"Impossible"<<"\n";
                    return 0;
                }
            }
            else if(i&&a[i]-a[i-1]>1){
                 cout<<"Impossible"<<"\n";
                    return 0;
            }
        }
    }else{
         int cnt=0;
        for(int i=0;i<n;i++){
            if(a[i]<=(d+1)/2){
                cnt++;
                if(a[i]<(d+1)/2){
                    cout<<"Impossible"<<"\n";
                    return 0;
                }
            }
            else if(i&&a[i]-a[i-1]>1){
                 cout<<"Impossible"<<"\n";
                    return 0;
            }
        }
        if(cnt<2){
            cout<<"Impossible"<<"\n";
                    return 0;
        }
    }
    cout<<"Possible"<<"\n";
}