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: Runtime Error)

import sys

def main():
    N = int(input())
    D = map(lambda i: int(i), input().split(' '))
    maximum = 0
    ary = [0] * (N - 1)
    for d in D:
        ary[d] = ary[d] + 1
        if maximum < d:
            maximum = d
    answer = 1
    for i in range(maximum + 1):
        if i == 0:
            if ary[i] != 1:
                print(0)
                sys.exit()
        elif ary[i] == 0:
            print(0)
            sys.exit()
        else:
            answer = (answer * (ary[i - 1] ** ary[i])) // 998244353
    print(answer)

main()