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 Rust (1.15.1) (judge verdict: Accepted)

use std::io::{stdin, Read, StdinLock};
use std::str::FromStr;
use std::iter::FromIterator;

fn main() {
    let cin = stdin();
    let mut scan = Scanner::new(cin.lock());
    let n = scan.read::<usize>();
    let a = scan.readn::<usize>(n);
    let diameter = a.iter().cloned().max().unwrap();
    let mut cnt = a.iter().cloned().collect::<Counter<_>>();
    'outer: loop {
        if diameter % 2 == 0 {
            for k in diameter / 2 + 1..diameter + 1 {
                if cnt[k] < 2 {
                    break 'outer;
                }
                cnt.dec(k, 2);
            }
            if cnt[diameter / 2] < 1 {
                break 'outer;
            } else {
                cnt.dec(diameter / 2, 1);
            }
        } else {
            for k in (diameter + 1) / 2..diameter + 1 {
                if cnt[k] < 2 {
                    break 'outer;
                }
                cnt.dec(k, 2);
            }
        }
        if cnt.keys().any(|&x| x < (diameter + 1)/2 + 1) {
            break 'outer;
        }
        println!("Possible");
        return;
    }
    println!("Impossible");
}

// region counter

use std::collections::HashMap;
use std::hash::Hash;
use std::ops::*;

#[derive(Clone, Debug)]
struct Counter<K: Eq + Hash> {
    cnt: HashMap<K, i64>,
    d: i64,
}

#[allow(dead_code)]
impl <K: Eq + Hash + Clone> Counter<K> {
    fn new() -> Counter<K> {
        Counter {cnt: HashMap::new(), d: 0}
    }
    fn dec(&mut self, key: K, delta: i64) {
        self[key.clone()] -= delta;
        if self[key.clone()] <= 0 {
            self.remove(&key);
        }
    }
}

#[allow(dead_code)]
impl <K: Eq + Hash> Deref for Counter<K> {
    type Target = HashMap<K, i64>;
    fn deref(&self) -> &Self::Target {
        &self.cnt
    }
}

#[allow(dead_code)]
impl <K: Eq + Hash> DerefMut for Counter<K> {
    fn deref_mut(&mut self) -> &mut Self::Target {
        &mut self.cnt
    }
}

#[allow(dead_code)]
impl<K: Eq + Hash> Index<K> for Counter<K> {
    type Output = i64;

    fn index(&self, index: K) -> &Self::Output {
        self.cnt.get(&index).unwrap_or(&self.d)
    }
}

#[allow(dead_code)]
impl<K: Eq + Hash + Clone> IndexMut<K> for Counter<K> {
    fn index_mut(&mut self, index: K) -> &mut i64 {
        if !self.cnt.contains_key(&index) {
            self.cnt.insert(index.clone(), self.d);
        }
        self.cnt.get_mut(&index).unwrap()
    }
}

#[allow(dead_code)]
impl<K: Eq + Hash> std::iter::FromIterator<K> for Counter<K> {
    fn from_iter<T: IntoIterator<Item=K>>(iter: T) -> Self {
        let mut cnt = HashMap::new();
        for i in iter {
            *cnt.entry(i).or_insert(0) += 1;
        }
        Counter {cnt: cnt, d: 0}
    }
}
// endregion

// region template
#[allow(dead_code)]
fn iu(i: i64) -> usize { i as usize }

#[allow(dead_code)]
fn ui(i: usize) -> i64 { i as i64 }

#[allow(dead_code, deprecated)]
fn join<T: std::fmt::Display>(slice: &[T], sep: &str) -> String {
    let strings = slice.iter().map(|t| format!("{}", t)).collect::<Vec<_>>();
    strings.connect(sep)
}

#[allow(dead_code)]
fn arr<'a, S, T>(n: usize, mut f: S) -> Vec<T> where S: FnMut(usize) -> T + 'a {
    (0..n).map(|i| f(i)).collect::<Vec<T>>()
}

#[allow(dead_code)]
fn alt<S, T>(v: Vec<S>) -> T where T: FromIterator<S> {
    v.into_iter().collect::<T>()
}

#[allow(dead_code)]
fn dup<S, T>(v: &[S]) -> T where T: FromIterator<S>, S: Clone {
    v.iter().cloned().collect::<T>()
}

struct Scanner<'a> {
    cin: StdinLock<'a>,
}

#[allow(dead_code)]
impl<'a> Scanner<'a> {
    fn new(cin: StdinLock<'a>) -> Scanner<'a> {
        Scanner { cin: cin }
    }

    fn read1<T: FromStr>(&mut self) -> Option<T> {
        let token = self.cin.by_ref().bytes().map(|c| c.unwrap() as char)
            .skip_while(|c| c.is_whitespace())
            .take_while(|c| !c.is_whitespace())
            .collect::<String>();
        token.parse::<T>().ok()
    }

    fn read<T: FromStr>(&mut self) -> T {
        self.read1().unwrap()
    }

    fn readn<T: FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read::<T>()).collect()
    }

    fn chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
}
// endregion