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: Accepted)

Z = 998244353


def calc(N, D):
    if D[0] != 0:
        return 0
    d = {}
    for Di in D:
        d[Di] = 1 + (d[Di] if Di in d else 0)
    if d[0] != 1:
        return 0
    result = 1
    for i in range(max(d.keys())):
        if not i + 1 in d:
            return 0
        result *= (d[i] ** d[i + 1]) % Z
        result = result % Z
    return result


N = int(input())
D = [int(s) for s in input().split()]
print(calc(N, D))