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 <iostream>
#include <iomanip>
#include <vector>
#include <algorithm>
#include <cstring>
#include <map>
#include <queue>
#include <cmath>
#include <complex> // complex<double> a(1.2 , 2.3);// real(): 1.2, imag()2.3
using namespace std;

#define MOD 998244353
#define EPS 1e-10
#define ll long long
#define ld long double
#define FOR(i,a,b) for(ll i=(ll)a;i<(ll)b;i++)
#define rep(i,n) FOR(i,0,n)
#define pb push_back
#define mp make_pair
#define all(a) a.begin(),a.end()
#define rall(a) a.rbegin(),a.rend()
#define rmsame(a) sort(all(a)),a.erase(unique(all(a)), a.end())
#define rmvector(a,b) rep(i,a.size())rep(j,b.size())if(a[i]==b[j]){a.erase(a.begin()+i);i--;break;}
#define first_more_idx(ve, num) upper_bound(all(ve), num)-ve.begin() // idx = first_more_idx(ve,num); if(idx = ve.size()) , then not exist
#define first_lessequal_idx(ve, num) lower_bound(all(ve), num)-ve.begin()
#define pq_pair_tB priority_queue <pair<ll,ll>, vector<pair<ll,ll> > , greater<pair<ll,ll> > >
#define pq_pair_ts priority_queue <pair<ll,ll> > // only sort first param
template<typename X> bool exist(vector<X> vec, X item){return find(all(vec), item)!=vec.end();}
ll gcd(ll a, ll b){if(b==0)return a;return gcd(b,a%b);}
ll lcm(ll a, ll b){return a/gcd(a,b)*b;}
ll power(ll x, ll n){if(n == 0) return 1;return  (n%2)? x*power(x, n-1)%MOD : power(x*x%MOD, n/2);}
ll dx[]={0, 0, -1, 1};
ll dy[]={1, -1, 0, 0};// up,down,left right

ll N ;
vector<ll> D;
ll dist[112345];
ll max_dist = 0 ;

int main(){
    cin.tie(0);
    ios::sync_with_stdio(false);

    cin >> N ;

    rep(i, N){
        ll inp; cin >>inp;
        D.pb(inp);
        max_dist = max(inp, max_dist);
        dist[inp] ++;
    }
    

    if(dist[0] != 1){
        cout << 0 << endl;
        return 0;
    }
    
    ll ans = 1 ;
    for(int i = 2; i <= max_dist; i++){
        ans *= power(dist[i-1], dist[i]) ;
        ans %= MOD;
    }

    cout << ans  << endl;
    
    //cout << fixed << setprecision(16) << ans << endl;
    return 0;
}