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)

from itertools import groupby
inputs = open(0).readlines()
n = int(inputs[0])
*D, = map(int, inputs[1].split())
D.sort()
mod = 998244353

keys, nodes = [], [] 
for k, g in groupby(D):
    keys.append(k)
    nodes.append(len(list(g)))

if not (keys[0] == 0 and nodes[0] == 1) or any(i != k for i, k in enumerate(keys)):
    print(0)
else:
    patterns = [1]
    for n0, n1 in zip(nodes, nodes[1:]):
        patterns.append(patterns[-1] * pow(n0, n1, mod) % mod)
    print(patterns[-1])