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++ (GCC 9.2.1) (judge verdict: Wrong Answer)

#include <bits/stdc++.h>
#define rep(i, n) for (int i = 0; i < (n); ++i)
using namespace std;
using ll = long long;
using P = pair<int, int>;

int const MOD = 998244353;

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

int main() {
    ll N;
    cin >> N;
    ll MAX = 0;
    vector<ll> D(N, 0);
    rep(i, N) {
        ll d;
        cin >> d;
        MAX = max(MAX, d);
        ++D[d];
    }

    ll ans = 1;
    rep(i, MAX + 1) {
        if (i == 0) {
            if (D[0] != 1) {
                ans = 0;
                break;
            } else
                continue;
        }

        if (D[i] == 0) {
            ans = 0;
            break;
        }

        ans = (ans * modpow(D[i - 1], D[i])) % MOD;
    }

    cout << ans << endl;
}