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)

/*
                ____________    ______________       __
               / _________  /\ /_____   _____/\     / /\
              / /\       / /  \\    /  /\    \ \   / /  \
             / /  \_____/ /   / \__/  /  \____\/  / /   /
            / /   /    / /   /    /  /   /       / /   /
           / /   /    / /   /    /  /   /       / /   /
          / /   /    / /   /    /  /   /       / /   /
         / /___/____/ /   /    /  /   /       / /___/________
        /____________/   /    /__/   /       /______________/\
        \            \  /     \  \  /        \              \ \
         \____________\/       \__\/          \______________\/
           ___       ___               ___    __________
          /  /\     /  /\             /  /\  /_______  /\
         /  /__\___/  /  \           /  /  \ \      /  /  \
        /____    ____/   /          /  /   /  \____/  /   /
        \   /   /\   \  /          /  /   /       /  /   /
         \_/   /  \___\/ ___      /  /   /       /  /   /
          /   /   /     /  /\    /  /   /       /  /   /
         /   /   /     /  /__\__/  /   /       /  /___/____
        /___/   /     /___________/   /       /___________/\
        \   \  /      \           \  /        \           \ \
         \___\/        \___________\/          \___________\/

          A FAN OF FIZZYDAVID

*/

#include<bits/stdc++.h>

#define HEAP priority_queue
#define rep(i,n) for(int i=0;i<(n);i++)
#define per(i,n) for(int i=(n)-1;i>=0;i--)
#define forn(i,l,r) for(int i=(l);i<=(r);i++)
#define nrof(i,r,l) for(int i=(r);i>=(l);i--)
#define mp make_pair
#define mt make_tuple
#define pb push_back
#define X first
#define Y second
#define eps 1e-20
#define pi 3.1415926535897932384626433832795
#define orz int
#define yjz main
#define fizzydavid return
#define ak 0
#define la ;
#define SZ(x) (int)x.size()
#define ALL(x) x.begin(),x.end()
#define FILL(a,b) memset((a),(b),sizeof((a)))

using namespace std;

typedef long long LL;
typedef double flt;
typedef vector<int> vi;
typedef pair<int,int> pii;
typedef vector<LL> vl;
typedef pair<int,LL> pil;
typedef pair<LL,int> pli;
typedef pair<LL,LL> pll;
typedef vector<pil> vil;
typedef vector<pii> vii;

const int iinf=1e9+7;
const LL linf=1ll<<60;
const flt dinf=1e10;

template <typename T>
inline void scf(T &x)
{
	bool f=0; x=0; char c=getchar();
	while((c<'0' || c>'9') && c!='-') c=getchar();
	if(c=='-') { f=1; c=getchar(); }
	while(c>='0' && c<='9') { x=x*10+c-'0'; c=getchar(); }
	if(f) x=-x; return;
}

template <typename T1,typename T2>
void scf(T1 &x,T2 &y) { scf(x); return scf(y); }

template <typename T1,typename T2,typename T3>
void scf(T1 &x,T2 &y,T3 &z) { scf(x); scf(y); return scf(z); }

template <typename T1,typename T2,typename T3,typename T4>
void scf(T1 &x,T2 &y,T3 &z,T4 &w) { scf(x); scf(y); scf(z); return scf(w); }

#ifdef ONLINE_JUDGE
#define debug(x) ;
#else
#define DEBUG
#define debug(x) cerr<<#x<<"="<<x<<endl;
#endif

//---------------------------head----------------------------

const int N=110;

int n,mx;
int a[N],cnt[N];

orz yjz()
{
    scf(n);

    forn(i,1,n)
    {
        scf(a[i]);
        cnt[a[i]]++;
        mx=max(mx,a[i]);
    }

    if(n==2)
    {
        if(cnt[1]!=2)
        {
            puts("Impossible");
        }
        else
            puts("Possible");
        fizzydavid ak la
    }

    if(n==3)
    {
        if(cnt[2]!=2 || cnt[1]!=1)
        {
            puts("Impossible");
        }
        else
            puts("Possible");
        fizzydavid ak la
    }

    if(mx==1)
    {
        puts("Impossible");
        fizzydavid ak la
    }

    if(mx==2)
    {
        if(cnt[2]!=n-1)
        {
            puts("Impossible");
        }
        else puts("Possible");
        fizzydavid ak la
    }

    forn(i,0,mx)
    {
        int d=max(i,mx-i);
        cnt[d]--;
    }

    forn(i,0,mx)
    {
        if(cnt[i]<0)
        {
            puts("Impossible");
            fizzydavid ak la
        }
        if(cnt[i] && i+i<mx)
        {
            puts("Impossible");
            fizzydavid ak la
        }
    }

    puts("Possible");
	fizzydavid ak la
}