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 PyPy3 (2.4.0) (judge verdict: Accepted)

import sys
input = sys.stdin.readline
from operator import itemgetter
sys.setrecursionlimit(10000000)
from collections import Counter


def main():
    n = int(input().strip())
    d = list(map(int, input().strip().split()))
    if d[0] != 0:
        print(0)
        return
    e = sorted(list(set(d)))
    for i, v in enumerate(e):
        if i != v:
            print(0)
            return
    v = dict(Counter(d))
    if v[0] != 1:
        print(0)
        return
    i = 0
    ans = 1
    while True:
        if i in v:
            if i == 0:
                i += 1
                continue
            ans *= v[i-1] ** v[i]
            ans %= 998244353
        else:
            break
        i += 1
    print(ans)

if __name__ == '__main__':
    main()