Problem: You are given a sequence of N integers: A_1,A_2,...,A_N.

Find the number of permutations p_1,p_2,...,p_N of 1,2,...,N that can be changed to A_1,A_2,...,A_N by performing the following operation some number of times (possibly zero), modulo 998244353:

* For each 1\leq i\leq N, let q_i=min(p_{i-1},p_{i}), where p_0=p_N. Replace the sequence p with the sequence q.

Constraints

* 1 \leq N \leq 3 × 10^5
* 1 \leq A_i \leq N
* All values in input are integers.

Input

Input is given from Standard Input in the following format:


N
A_1
:
A_N


Output

Print the number of the sequences that satisfy the condition, modulo 998244353.

Examples

Input

3
1
2
1


Output

2


Input

5
3
1
4
1
5


Output

0


Input

8
4
4
4
1
1
1
2
2


Output

24


Input

6
1
1
6
2
2
2


Output

0

Solution (in CPP):

Solution:
#include <string>
#include <vector>
#include<iostream>
#include<cstdio>
#include<cstdlib>
#include<stack>
#include<queue>
#include<cmath>
#include<algorithm>
#include<functional>
#include<list>
#include<deque>
#include<bitset>
#include<set>
#include<map>
#include<unordered_map>
#include<unordered_set>
#include<cstring>
#include<sstream>
#include<complex>
#include<iomanip>
#include<numeric>
#include<cassert>
#define X first
#define Y second
#define pb push_back
#define rep(X,Y) for (int (X) = 0;(X) < (Y);++(X))
#define reps(X,S,Y) for (int (X) = S;(X) < (Y);++(X))
#define rrep(X,Y) for (int (X) = (Y)-1;(X) >=0;--(X))
#define rreps(X,S,Y) for (int (X) = (Y)-1;(X) >= (S);--(X))
#define repe(X,Y) for ((X) = 0;(X) < (Y);++(X))
#define peat(X,Y) for (;(X) < (Y);++(X))
#define all(X) (X).begin(),(X).end()
#define rall(X) (X).rbegin(),(X).rend()
#define eb emplace_back
#define UNIQUE(X) (X).erase(unique(all(X)),(X).end())
#define Endl endl

using namespace std;
typedef long long ll;
typedef pair<int,int> pii;
typedef pair<ll,ll> pll;
template<class T> using vv=vector<vector<T>>;
template<class T> ostream& operator<<(ostream &os, const vector<T> &t) {
os<<"{"; rep(i,t.size()) {os<<t[i]<<",";} os<<"}"<<endl; return os;}
template<class T,size_t n> ostream& operator<<(ostream &os, const array<T,n> &t) {
	os<<"{"; rep(i,n) {os<<t[i]<<",";} os<<"}"<<endl; return os;}
template<class S, class T> ostream& operator<<(ostream &os, const pair<S,T> &t) { return os<<"("<<t.first<<","<<t.second<<")";}
template<class S, class T,class U> ostream& operator<<(ostream &os, const tuple<S,T,U> &t) { return os<<"("<<get<0>(t)<<","<<get<1>(t)<<","<<get<2>(t)<<")";}
template<class S, class T,class U,class V> ostream& operator<<(ostream &os, const tuple<S,T,U,V> &t) { return os<<"("<<get<0>(t)<<","<<get<1>(t)<<","<<get<2>(t)<<","<<get<3>(t)<<")";}
template<class S, class T,class U,class V,class W> ostream& operator<<(ostream &os, const tuple<S,T,U,V,W> &t) { return os<<"("<<get<0>(t)<<","<<get<1>(t)<<","<<get<2>(t)<<","<<get<3>(t)<<","<<get<4>(t)<<")";}
template<class T> inline bool MX(T &l,const T &r){return l<r?l=r,1:0;}
template<class T> inline bool MN(T &l,const T &r){return l>r?l=r,1:0;}
//#undef NUIP
#ifdef NUIP
#define out(args...){vector<string> a_r_g_s=s_p_l_i_t(#args, ','); e_r_r(a_r_g_s.begin(), args); }
vector<string> s_p_l_i_t(const string &s, char c){vector<string> v;int d=0,f=0;string t;for(char c:s){if(!d&&c==',')v.pb(t),t="";else t+=c;if(c=='\"'||c=='\'')f^=1;if(!f&&c=='(')++d;if(!f&&c==')')--d;}v.pb(t);return move(v);}
void e_r_r(vector<string>::iterator it) {}
template<typename T, typename... Args> void e_r_r(vector<string>::iterator it, T a, Args... args){ if(*it==" 1"||*it=="1") cerr<<endl; else cerr << it -> substr((*it)[0] == ' ', it -> length()) << " = " << a << ", "; e_r_r(++it, args...);}
#else
#define out
#endif
#ifdef __cpp_init_captures
template<typename T>vector<T> table(int n, T v){ return vector<T>(n, v);}
template <class... Args> auto table(int n, Args... args){auto val = table(args...); return vector<decltype(val)>(n, move(val));}
#endif
const ll MOD=998244353;

ll modpow(ll r,ll n,ll m=MOD){
  ll re=1,d=r%m;
  if(n<0)(n%=m-1)+=m-1;
  for(;n;n/=2){
    if(n&1)(re*=d)%=m;
    (d*=d)%=m;
  }
  return re;
}
template <int mod=MOD> struct modInt{
  int v;
  modInt(int v=0):v(v){}
  modInt operator+(const modInt &n)const{return v+n.v<mod ? v+n.v : v+n.v-mod;}
  modInt operator-(const modInt &n)const{return v-n.v<0 ? v-n.v+mod : v-n.v;}
  modInt operator*(const modInt &n)const{return ll(v)*n.v%mod;}
  modInt operator/(const modInt &n)const{return ll(v)*modpow(n.v%mod,-1,mod)%mod;}
  modInt operator+(const ll &n)const{return v+n<mod ? v+n : v+n-mod;}
  modInt operator-(const ll &n)const{return v-n<0 ? v-n+mod : v-n;}
  modInt operator*(const ll &n)const{return ll(v)*(n%mod)%mod;}
  modInt operator/(const ll &n)const{return ll(v)*modpow(n%mod,-1,mod)%mod;}
  modInt& operator+=(const modInt &n){v+=n.v; if(v>=mod) v-=mod; return *this;}
  modInt& operator-=(const modInt &n){v-=n.v; if(v<0) v+=mod; return *this;}
  modInt& operator*=(const modInt &n){v=ll(v)*n.v%mod; return *this;}
  modInt& operator/=(const modInt &n){v=ll(v)*modpow(n.v,-1,mod)%mod; return *this;}
  modInt& operator+=(const ll &n){v+=n; if(v>=mod) v-=mod; return *this;}
  modInt& operator-=(const ll &n){v-=n; if(v<0) v+=mod; return *this;}
  modInt& operator*=(const ll &n){v=ll(v)*n%mod; return *this;}
  modInt& operator/=(const ll &n){v=ll(v)*modpow(n,-1,mod)%mod; return *this;}
};
template<int mod> ostream& operator<<(ostream &os,const modInt<mod> &n){return os<<n.v;};
template<int mod> modInt<mod> operator+(const ll &n,const modInt<mod> &m){return m.v+n<mod ? m.v+n : m.v+n-mod;}
template<int mod> modInt<mod> operator-(const ll &n,const modInt<mod> &m){return n-m.v<0 ? n-m.v+mod : n-m.v;}
template<int mod> modInt<mod> operator*(const ll &n,const modInt<mod> &m){return ll(m.v)*(n%mod)%mod;}
template<int mod> modInt<mod> operator/(const ll &n,const modInt<mod> &m){return ll(m.v)*modpow(n%mod,-1,mod)%mod;}
typedef modInt<MOD> mint;
template <int mod> modInt<mod> modpow(modInt<mod> r,ll n){ modInt<mod> re(1); if(n<0)(n%=mod-1)+=mod-1; for(;n;n/=2){if(n&1) re*=r; r*=r;} return re;}
vector<mint> fact,finv,inv;
mint comb(ll n,ll r){ if(n<r||r<0)return 0; return fact[n]*finv[n-r]*finv[r];}
class Doralion{
  void Modinvs(vector<mint> &re,int n){ re.resize(n+1); re[1]=1; for(int i=2;i<=n;++i)re[i]=re[MOD%i]*(MOD-MOD/i);}
  void Facts(vector<mint> &re,int n){ re.resize(n+1); re[0]=1; rep(i,n)re[i+1]=re[i]*(i+1);}
  void Factinvs(vector<mint> &re,const vector<mint> &inv,int n){ re.resize(n+1); re[0]=1; rep(i,n)re[i+1]=re[i]*inv[i+1];}
public:
  Doralion(int n){ Modinvs(inv,n); Facts(fact,n); Factinvs(finv,inv,n);}
} doralion(512345);

int main(){
  ios_base::sync_with_stdio(false);
  cout<<fixed<<setprecision(0);
	int n;
	cin>>n;
	vector<int> a(n);
	rep(i,n) cin>>a[i], --a[i];
	vector<int> cnt(n);
	for(int x:a) ++cnt[x];
	if(*max_element(all(cnt))==n){
		if(a[0]){
			cout<<0<<endl;
		}else{
			cout<<fact[n]<<endl;
		}
		return 0;
	}
	{
		int mx=*max_element(all(a));
		rep(i,n-1)if(a[i]==mx && a[i]!=a[i+1]){
			vector<int> v;
			reps(j,i+1,n) v.pb(a[j]);
			rep(j,i+1) v.pb(a[j]);
			a=v;
			break;
		}
	}
	int t=cnt[0];
	vector<pii> ints(n,pii(MOD,0));
	rep(i,n) ints[a[i]].Y=i;
	rrep(i,n) ints[a[i]].X=i;
	auto no=[](){ cout<<0<<endl; exit(0); };
	rep(i,n)if(cnt[i] && ints[i].Y-ints[i].X+1!=cnt[i]) no();
	mint re=1;
	int fr=0;
	set<int> st;
	reps(i,-1,n+1) st.emplace(i);
	out(a,1);
	rep(i,n){
		out(i,fr,re,1);
		if(!cnt[i]){
			if(fr<=i) no();
			re*=fr-i;
			continue;
		}
		if(cnt[i]>t) no();
		if(cnt[i]==t){
		}
		if(cnt[i]<t){
			int l,r;
			tie(l,r)=ints[i];
			int L=0,R=0;
			if(a[(l+n-1)%n]<i) L=1;
			if(a[(r+1)%n]<i) R=1;
			if(!L && !R) no();
			if(L&&R){
				re*=t+1-cnt[i];
			}else{
			}
		}
		reps(j,ints[i].X,ints[i].Y+1) st.erase(j);
		auto it=st.lower_bound(ints[i].X); --it;
		int l=*it;
		++it;
		int r=*it;
		fr-=max(0,ints[i].X-l-1-t+1);
		fr-=max(0,r-ints[i].Y-1-t+1);
		fr+=max(0,r-l-1-t+1);
		if(l==-1 && r==n) fr=n;
		// for(int x:st) cout<<x<<",";cout<<endl;
	}
	cout<<re<<endl;
  return 0;
}