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 _GLIBCXX_DEBUG
#define rep(i,n) for(ll i=0;i< (n);i++)
#define rep1(i,n) for(ll i=1;i<= (n);i++)
using namespace std;
typedef long long ll;
typedef pair<ll,ll> P;

const static ll MOD = 998244353;

ll modcross(ll a,ll n){
    if(n == 0)return 0;
    if(n % 2 == 1)return a+modcross(a,n-1) % MOD;
    ll t = modcross(a,n/2) % MOD;
    return t*2 % MOD;
}

ll modpow(ll a,ll n){
    if(n == 0)return 1;
    if(n % 2 == 1)return a*modpow(a,n-1) % MOD;
    ll t = modpow(a,n/2) % MOD;
    return t*t % MOD;
}

int main(){
    ll n;cin>>n;
    vector<ll> D(n,0);
    rep(i,n)cin>>D[i];
    map<ll,ll> mp;
    rep(i,n){
        mp[D[i]]++;
    }
    if(D[0] != 0 || mp[0] != 1){
        cout<<'0'<<endl;
        return 0;
    }
    ll ans = 0;
    for(auto it = mp.begin();it!=mp.end();it++){
        if(it == mp.begin())ans++;
        else{
            if(prev(it,1)->second != 1 ){
                ans = modcross(ans,modpow(prev(it,1)->second,it->second));
            }
        }
    }
    cout<<ans%MOD<<endl;
    return 0;
}