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
N = int(input())
d_list = [int(d) for d in input().split(' ')]


def search(d_list):
    if d_list[0] != 0:
        return 0

    d_dict = defaultdict(int)
    unique_values = set()
    for d in d_list:
        d_dict[d] += 1
        unique_values.add(d)

    unique_num = len(unique_values)
    if not sum(unique_values) == unique_num * (unique_num - 1) / 2:
        return 0

    result = 1

    for v in unique_values:
        if v == 0:
            continue
        prev_node_num = d_dict.get(v-1)
        current_node_num = d_dict.get(v)

        result *= prev_node_num ** current_node_num
    return result % 998244353

print(search(d_list))