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: Accepted)

#include <bits/stdc++.h>
using namespace std;
#define int long long
#define in ({int x=0;int c=getchar(),n=0;for(;!isdigit(c);c=getchar()) n=(c=='-');for(;isdigit(c);c=getchar()) x=x*10+c-'0';n?-x:x;})
mt19937 rng(chrono::steady_clock::now().time_since_epoch().count());
int rnd(int l,int r){return l+rng()%(r-l+1);}
#define fasty ios_base::sync_with_stdio(false),cin.tie(nullptr);
#define task "TASK"
#define forinc(a,b,c) for(int a=b,_c=c;a<=_c;++a)
#define fordec(a,b,c) for(int a=b,_c=c;a>=_c;--a)
#define forv(a,b) for(auto&a:b)
#define fi first
#define se second
#define pb push_back
#define ii pair<int,int>
#define mt make_tuple
#define all(a) a.begin(),a.end()
#define reset(f, x) memset(f, x, sizeof(f))
#define bit(x,i) ((x>>(i-1))&1)
#define on(x,i) (x|(1ll<<(i-1)))
#define off(x,i) (x&~(1<<(i-1)))

const int N=100010;

int pw(int a,int i){
    int b=1;
    for(;i;i>>=1,a=a*a%998244353) if(i&1) b=b*a%998244353;
    return b;
}

main(){
    //freopen(task".inp","r",stdin);
    //freopen(task".out","w",stdout);
    int n; n=in; vector<int> cnt(n),a(n+1),g(n);
    forinc(i,1,n) cnt[a[i]=in]++;
    int mx=*max_element(all(a));
    if(a[1] || cnt[0]>1) return cout<<0,0;
    forinc(i,1,mx) if(!cnt[i]) return cout<<0,0;
    long long ret=1;
    forinc(i,1,mx){
        ret=ret*pw(cnt[i-1],cnt[i])%998244353;
    }
    cout<<ret;
}