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)

N = int(input())
D = map(int, input().split())

r = [None] * N
m = 0

for i, j in enumerate(D):
    if i == 0 and j != 0:
        print("0")
        exit()
    r[j] = 1 if r[j] is None else r[j] + 1
    if j > m:
        m = j

result = 1
pre = None

for i, x in enumerate(r):
    if x is None:
        if i != m + 1:
            print("0")
            exit()
        else:
            break
    if i < 2:
        pre = x
        continue
    result *= pre**x
    pre = x


print(result)