Problem:
Score : 300 points

Problem Statement

Given is an integer sequence D_1,...,D_N of N elements. Find the number, modulo 998244353, of trees with N vertices numbered 1 to N that satisfy the following condition:

For every integer i from 1 to N, the distance between Vertex 1 and Vertex i is D_i.

Notes

A tree of N vertices is a connected undirected graph with N vertices and N-1 edges, and the distance between two vertices are the number of edges in the shortest path between them.

Two trees are considered different if and only if there are two vertices x and y such that there is an edge between x and y in one of those trees and not in the other.

Constraints

1 \leq N \leq 10^5

0 \leq D_i \leq N-1

Input

Input is given from Standard Input in the following format:

N
D_1 D_2 ... D_N

Output

Print the answer.

Sample Input 1

4
0 1 1 2

Sample Output 1

2

For example, a tree with edges (1,2), (1,3), and (2,4) satisfies the condition.

Sample Input 2

4
1 1 1 1

Sample Output 2

0

Sample Input 3

7
0 3 2 1 2 2 1

Sample Output 3

24

Solution in Rust (1.15.1) (judge verdict: Wrong Answer)

fn read<T: std::str::FromStr>() -> T {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).ok();
    s.trim().parse().ok().unwrap()
}

fn read_vec<T: std::str::FromStr>() -> Vec<T> {
    read::<String>()
        .split_whitespace()
        .map(|e| e.parse().ok().unwrap())
        .collect()
}

fn read_touple<T: std::str::FromStr + Copy>() -> (T, T) {
    let v: Vec<T> = read_vec();
    assert_eq!(v.len(), 2);
    (v[0], v[1])
}

const MOD: usize = 998244353;
fn main() {
    let n: usize = read();
    let mut d: Vec<usize> = read_vec();
    d.sort();
    let v = split_same_element(&d);
    let m = v.len();
    let (_, mut tot) = v[0];
    let mut ans = 1;
    for (i, &e) in v.iter().enumerate() {
        let (idx, _) = e;
        if idx != i {
            println!("{}", 0);
            return;
        }
    }

    if tot != 1 {
        println!("{}", 0);
        return;
    }

    for i in 1..m {
        let (_, t) = v[i - 1];
        let (_, tot) = v[i];
        ans *= pow(t, tot);
        ans %= MOD;
    }

    println!("{}", ans);
}

fn pow(a: usize, n: usize) -> usize {
    let mut res = 1;
    let mut a = a;
    let mut n = n;

    while n > 0 {
        if n % 2 == 0 {
            a = a * a;
            n /= 2;
        } else {
            res = res * a;
            a = a * a;
            n /= 2;
        }
        res %= MOD;
        a %= MOD;
    }
    res
}

fn split_same_element<T: Eq + Copy>(v: &Vec<T>) -> Vec<(T, usize)> {
    let mut res: Vec<(T, usize)> = Vec::new();
    if v.len() == 0 {
        return res;
    }

    let mut now: T = v[0];
    let mut count: usize = 0;
    for &e in v {
        if e == now {
            count += 1;
        } else {
            res.push((now, count));
            count = 1;
            now = e;
        }
    }
    res.push((now, count));

    res
}