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 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include<bits/stdc++.h>
using namespace std;

#define i64 long long 

i64 POW(i64 x,i64 n, i64 MOD = 1000000007){
    i64 res = 1;
    while(n){
        if(n%2) ((res *= x) %= MOD);
        ((x *= x) %= MOD);
        n >>= 1;
    }
    return res;
}

int main(){
    int N;
    cin >> N;
    vector<int> D(N);
    for(int i = 0; i < N; i++) cin >> D[i];

    if(D[0] != 0){
        cout << 0 << endl;
        return 0;
    }

    vector<long long> d(N, 0);
    for(int i = 0; i < N; i++) d[ D[i] ]++;
    if(d[0] != 1){
        cout << 0 << endl;
        return 0;
    }

    int zero = 0;
    for(int i = 1; d[i] != 0; i++) zero = i+1;
    for(int i = 1; i < N; i++){
        if(d[i] != 0 && zero < i){
            cout << 0 << endl;
            return 0;
        }
    }

    const long long MOD = 998244353;
    long long ans = 1;
    for(int i = 2; i < zero; i++){
        ans = (ans*POW(d[i-1], d[i])) % MOD;
    }

    cout << ans << endl;
}