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 PHP7 (7.0.15) (judge verdict: Wrong Answer)

<?php
    $f = true;
	$n = trim(fgets(STDIN));
	$a = explode(" ", trim(fgets(STDIN)));
	rsort($a);
	for($i=1; $i<=$n; $i++) $c[$i] = 0;
	for($i=0; $i<$n; $i++){
	    $x = $a[$i];
	    $c[$x]++;
	}
	
	$tmp = $a[0];
	$r = 1;
	for($i=0; $i<$n; $i++){
	    if(!$f) break;
	    $x = $a[$i];
	    if($x != $tmp){
	        if($x != $tmp - 1) $f = false;
	        else{
	            $r++;
    	        if($c[$tmp] < $c[$x]) $f = false;
    	        $tmp = $x;
	        }
	    }
	}
	
	for($i=0; $i<$n; $i++){
	    $x = $a[$i];
	    if($x == $tmp) break;
	    if($c[$x] < 2) $f = false;
	}
	
	if($f){
	    if($c[$tmp] != 1){
	        if($tmp != $r) $f = false;
	    } else {
	        if($tmp != $r - 1) $f = false;
	    }
	}
	
	if($f) echo "Possible";
	else echo "Impossible";
?>