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)

# メモ
# 連結無効グラフ　＝　全部が繋がっている

# N = int(input())
# D = list(map(int, input()))

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

JOSU = 998244353

anser = 0

deep = [-1] * (N + 1)
deep[0] = 0
memo = [0] * N

# 各深さの数を求める
for i in range(N):
    if deep[D[i]] == -1:
        deep[D[i]] = 0
    deep[D[i]] += 1

def dfs(n, end, sum):
    if n == end:
        return sum
    elif deep[n] == -1 or deep[n + 1] == -1:
        return dfs(n + 1, end, sum)
    else:
        return dfs(n + 1, end, sum * deep[n] ** deep[n + 1])

print(dfs(0, N, 1) % JOSU)