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: Accepted)

#![allow(unused)]
use std::io::Write;

#[allow(unused)]
macro_rules! debug {
    ($($format:tt)*) => (write!(std::io::stderr(), $($format)*).unwrap());
}
#[allow(unused)]
macro_rules! debugln {
    ($($format:tt)*) => (writeln!(std::io::stderr(), $($format)*).unwrap());
}

const M: usize = 998244353;

fn main() {
    let n: usize = {
        let mut s = String::new();
        std::io::stdin().read_line(&mut s).unwrap();
        let s = s.trim_right().to_owned();
        s.parse().unwrap()
    };

    let d: Vec<usize> = {
        let mut s = String::new();
        std::io::stdin().read_line(&mut s).unwrap();
        let s = s.trim_right().to_owned();
        s.split_whitespace().map(|x| x.parse().unwrap()).collect()
    };

    if d[0] != 0 {
        println!("{}", 0);
        return;
    }

    let mut e = vec![0; n];
    let mut max_e = 0;
    for a in d {
        e[a] += 1;
        if a > max_e {
            max_e = a;
        }
    }

    // debugln!("{:?}", e);
    // debugln!("{}", max_e);

    if e[0] != 1 {
        println!("{}", 0);
        return;
    }

    let mut res = 1;
    for i in 1..(max_e + 1) {
        res *= pow_mod(e[i - 1], e[i]);
        res %= M;
    }

    println!("{}", res);

    // debugln!("{}", pow_mod(0, 0));
    // debugln!("{}", pow_mod(0, 1));
    // debugln!("{}", pow_mod(0, 2));
    // debugln!("{}", pow_mod(0, 3));
    // debugln!("{}", pow_mod(10, 0));
    // debugln!("{}", pow_mod(3, 1));
    // debugln!("{}", pow_mod(2, 2));
    // debugln!("{}", pow_mod(2, 3));
    // debugln!("{}", pow_mod(2, 10000000000000000000));
}

fn pow_mod(a: usize, b: usize) -> usize {
    if a == 0 {
        return 0;
    }
    if b == 0 {
        return 1;
    }

    if b % 2 == 0 {
        pow_mod(a, b / 2).pow(2) % M
    } else {
        (pow_mod(a, b - 1) * a) % M
    }
}