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 PyPy3 (2.4.0) (judge verdict: Wrong Answer)

import sys

sys.setrecursionlimit(4100000)


def inputs(num_of_input):
    ins = [input() for i in range(num_of_input)]
    return ins


def solve(inputs):
    SSSS = 998244353
    D = string_to_int(inputs[0])
    # {distance: numbers}
    nodes = {}
    for i, d in enumerate(D):
        i += 1
        if d not in nodes:
            nodes[d] = [i]
        else:
            nodes[d].append(i)

    if 0 not in nodes:
        return 0

    prev = -1
    prev_num = -1
    count = 0
    for d in sorted(nodes.keys()):
        if d == 0:
            if not (len(nodes[d]) == 1 and nodes[d][0] == 1):
                return 0
            count = 1
            prev_num = 1
            continue
        if abs(d - prev) == 1:
            return 0

        count *= prev_num ** len(nodes[d])
        prev_num = len(nodes[d])
    return count


def string_to_int(string):
    return list(map(lambda x: int(x), string.split()))


if __name__ == "__main__":
    input()
    ret = solve(inputs(1))
    print(ret)