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;
typedef long long ll;
const static ll INF = 1e9;
//const static ll MOD = 1e9+7;

long long modpow(long long a, long long n, long long mod) {
    long long res = 1;
    while (n > 0) {
        if (n & 1) res = res * a % mod;
        a = (a * a) % mod;
        n >>= 1;
    }
    return res;
}

const static ll mod = 998244353;
int main(){
    ll N; cin >> N;
    map<ll, ll> M;
    for(int i = 0; i < N; i++){
        ll x; cin >> x; M[x]++;
    }
    ll ans = 1;
    ll cachef=0;
    ll caches = M[0];

    if(caches == 0) {cout << 0 << endl; return 0;}
    for(auto m : M){
        if(m.first==0)continue;
        //cout << cachef << " " << m.first << endl;
        if(m.first-cachef>1){
            cout << 0 << endl; return 0;
        }
        ll temp = modpow(caches, m.second, mod)%mod;
        ans*=temp;
        ans %= mod;
        caches = m.second;
        cachef = m.first;
    }
    cout << ans%mod << endl;


}