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 Python (3.4.3) (judge verdict: Wrong Answer)

from collections import defaultdict


def main():
    n = int(input())
    d_s = list(map(int, input().split(' ')))

    if d_s[0] != 0:
        print(0)
        quit()

    dic = defaultdict(int)

    for d in d_s:
        dic[d] += 1

    t_s = sorted(dic.items())

    ans = 1

    pre_n = -1
    pre_count = 1


    for t in t_s:
        if t[0] - 1 != pre_n:
            print(0)
            quit()
        ans *= pre_count ** t[1]
        pre_n += 1
        pre_count = t[1]
    print(ans % 998244353)


if __name__ == '__main__':
    main()