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 (1.42.0) (judge verdict: Accepted)

fn main() {
    proconio::input! {
        n: usize,
        ds: [usize; n],
    }
    const M: usize = 998244353;
    if ds[0] != 0 || ds.iter().skip(1).any(|d| *d == 0) {
        println!("0");
        return;
    }
    let mut count = 1usize;
    let mut m = std::collections::BTreeMap::new();
    for d in ds.into_iter().skip(1) {
        *m.entry(d).or_default() += 1;
    }
    let mut distance = 0;
    let mut prev = 1usize;
    for (d, num) in m.iter() {
        if distance + 1 != *d {
            println!("0");
            return;
        }
        for _ in 0..*num {
            count *= prev;
            count %= M;
        }
        distance = *d;
        prev = *num;
    }
    println!("{}", count);
}