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 C++14 (Clang 3.8.0) (judge verdict: Wrong Answer)

#include <iostream>
#include <cmath>
#include <vector>

using std::cin;
using std::cout;
using std::endl;

int main() {
    int N;
    cin >> N;

    std::vector<long long> D(N), T(N);
    int maxT = 0;

    for (int i = 0; i < N; i++) {
        cin >> D.at(i);
        if (i == 0 && D.at(0) != 0) {
            cout << 0 << endl;
            return 0;
        }
        if (i != 0 && D.at(i) == 0) {
            cout << 0 << endl;
            return 0;
        }
        if (D.at(i) > maxT) {
            maxT = static_cast<int>(D.at(i));
        }

        T.at(D.at(i))++;
    }

    long long count = 1;
    for (int i = 1; i < maxT; i++) {
        if (T.at(i) == 0) {
            cout << 0 << endl;
            return 0;
        }
        count *= std::pow(T.at(i), T.at(i + 1));
    }
    cout << count % 998244353 << endl;
    return 0;
}