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>
#define REP(i,n) for(int (i)=0;(i)<(n);(i)++)
using namespace std;
typedef long long ll;
const ll mod = 1e9+7;
const ll modu = 998244353;
#define all(x) (x).begin(),(x).end()

template<class T> inline void chmin(T& a, T b) {
    if (a > b) {
        a = b;
    }
}

template<class T> inline void chmax(T& a, T b) {
    if (a < b) {
        a = b;
    }
}

int main(){
    cin.tie(0);
    ios::sync_with_stdio(false);
    std::cout<<std::fixed<<std::setprecision(10);
    int N;
    cin >> N;
    vector<int> D(N),E(N);
    REP(i,N)cin >> D[i];
    int maxis = 0;
    REP(i,N){
        E[D[i]]++;
        maxis = max(maxis,D[i]);
    }
    ll match = 1;
    REP(i,maxis+1){
        if(i == 0){
            if(E[0] != 1)match = 0;
        }else{
            ll pow = 1;
            REP(l,E[i]){
                pow*=E[i-1];
                pow%=modu;
            }
            match*=pow;
        }
    }
    cout <<match<< "\n";
}