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)

N = int(input())
A = list(map(int, input().split()))
k = max(A)

def pow_k(x, n):
    """
    O(log n)
    """
    if n == 0:
        return 1

    K = 1
    while n > 1:
        if n % 2 != 0:
            K *= x % 998244353
        x *= x % 998244353
        n //= 2

    return K * x % 998244353

if A[0] != 0:
    print(0)
    exit()
if A.count(0) != 1:
    print(0)
    exit()



R = [1]
result = 1


for i in range(k):
    R.append(A.count(i + 1))
    A.remove(i + 1)

for j in range(k):
    result *= pow_k(R[j], R[j + 1])

    if result == 0:
        print(0)
        exit()

print(result % 998244353)