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 ll INF = 1e17;
// https://atcoder.jp/contests/abc143/submissions/8034843

ll const MOD = 998244353;

// modをとりながらpowする
ll modpow( ll a, ll b)
{
    ll layer=1;
    for (ll i=0; i< b; i++)
    {
        layer *= a;
        layer %= MOD;
    }
    return layer;
}

int main() {
    // input
    ll N;
    cin >> N;
    map<ll,ll> map_D;
    for (ll i=0; i < N; i++)
    {
        ll d;
        cin >> d;
        map_D[d]++;
    }

    // 頂点は一か所でないとダメ
    if (map_D[0] !=1)
    {
        cout << 0 << endl;
        return 0;
    }

    // resolve
    ll index=0;
    ll before_num=1;
    ll ans = 1;
    for(auto d: map_D)
    {
        if (index != d.first)
        {
            cout << 0 << endl;
            return 0;
        }     
        ans *= modpow(before_num, d.second);
        ans %= MOD;
        before_num = d.second;
        index++;
    }
    cout << ans << endl;

    return 0;
}