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: Time Limit Exceeded)

N = int(input())
D = list(map(int, input().split()))
L = list()
result = 1
m = max(D)+1

def pow_r(x, n):
  if n == 0:
    return 1
  if n % 2 == 0:
    return pow_r(x ** 2, n // 2)
  else:
    return x * pow_r(x ** 2, (n-1) // 2)

for i in range(1, m):
  L.append(D.count(i))

for j in range(1,len(L)):
  #result *= L[j-1] ** L[j]
  result *= pow_r(L[j-1], L[j])

if D.count(0) == 1 and D[0] == 0:
  print(result % 998244353)
else:
  print(0)