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 <bits/stdc++.h>
using namespace std;

using ll=int64_t;
#define int ll

#define FOR(i,a,b) for(int i=int(a);i<int(b);i++)
#define REP(i,b) FOR(i,0,b)
#define MP make_pair
#define PB push_back
#define ALL(x) x.begin(),x.end()
#ifdef LOCAL
#define cerr (cerr<<"-- line "<<__LINE__<<" -- ")
#else
class CerrDummy{}cerrDummy;
template<class T>
CerrDummy& operator<<(CerrDummy&cd,const T&){
	return cd;
}
using charTDummy=char;
using traitsDummy=char_traits<charTDummy>;
CerrDummy& operator<<(CerrDummy&cd,basic_ostream<charTDummy,traitsDummy>&(basic_ostream<charTDummy,traitsDummy>&)){
	return cd;
}
#define cerr cerrDummy
#endif
#define REACH cerr<<"reached"<<endl
#define DMP(x) cerr<<#x<<":"<<x<<endl
#define ZERO(x) memset(x,0,sizeof(x))
#define ONE(x) memset(x,-1,sizeof(x))

using pi=pair<int,int>;
using vi=vector<int>;
using ld=long double;

template<class T,class U>
ostream& operator<<(ostream& os,const pair<T,U>& p){
	os<<"("<<p.first<<","<<p.second<<")";
	return os;
}

template<class T>
ostream& operator <<(ostream& os,const vector<T>& v){
	os<<"{";
	REP(i,(int)v.size()){
		if(i)os<<",";
		os<<v[i];
	}
	os<<"}";
	return os;
}

ll read(){
	ll i;
	scanf("%"  SCNd64,&i);
	return i;
}

void printSpace(){
	printf(" ");
}

void printEoln(){
	printf("\n");
}

void print(ll x,int suc=1){
	printf("%" PRId64,x);
	if(suc==1)
		printEoln();
	if(suc==2)
		printSpace();
}

string readString(){
	static char buf[3341000];
	scanf("%s",buf);
	return string(buf);
}

char* readCharArray(){
	static char buf[3341000];
	static int bufUsed=0;
	char* ret=buf+bufUsed;
	scanf("%s",ret);
	bufUsed+=strlen(ret)+1;
	return ret;
}

template<class T,class U>
void chmax(T& a,U b){
	if(a<b)
		a=b;
}

template<class T,class U>
void chmin(T& a,U b){
	if(b<a)
		a=b;
}

template<class T>
T Sq(const T& t){
	return t*t;
}

#define CAPITAL
void Yes(bool ex=true){
	#ifdef CAPITAL
	cout<<"YES"<<endl;
	#else
	cout<<"Yes"<<endl;
	#endif
	if(ex)exit(0);
}
void No(bool ex=true){
	#ifdef CAPITAL
	cout<<"NO"<<endl;
	#else
	cout<<"No"<<endl;
	#endif
	if(ex)exit(0);
}

const ll infLL=LLONG_MAX/3;

#ifdef int
const int inf=infLL;
#else
const int inf=INT_MAX/2-100;
#endif

const int mod=998244353;
template<class T,class U>
void add(T& a,U b){
	//a=((ll)a+b)%mod;
	a+=b;
	if(a>=mod)a-=mod;
}

template<class T,class U>
void sub(T& a,U b){
	//a=((ll)a-b%mod+mod)%mod;
	a-=b;
	if(a<0)a+=mod;
}

template<class T,class U>
void mult(T& a,U b){
	a=((ll)a*b)%mod;
}

ll modPow(ll a,ll p){
	ll s=1;
	while(p){
		if(p&1)mult(s,a);
		mult(a,a);
		p>>=1;
	}
	return s;
}

ll modInv(ll a){
	return modPow(a,mod-2);
}

const int Vmax=1000010;
int fact[Vmax],factInv[Vmax],invs[Vmax];
void InitFact(){
	fact[0]=1;
	FOR(i,1,Vmax){
		fact[i]=fact[i-1];
		mult(fact[i],i);
	}
	factInv[Vmax-1]=modInv(fact[Vmax-1]);
	for(int i=Vmax-2;i>=0;i--){
		factInv[i]=factInv[i+1];
		mult(factInv[i],i+1);
	}
	for(int i=Vmax-1;i>=1;i--){
		invs[i]=factInv[i];
		mult(invs[i],fact[i-1]);
	}
}
int Choose(int n,int k){
	return ll(fact[n])*factInv[n-k]%mod*factInv[k]%mod;
}
int Binom(int a,int b){
	return ll(fact[a+b])*factInv[a]%mod*factInv[b]%mod;
}

void Impossible(){
	print(0);
	exit(0);
}

struct SlideMax{
	vi buf;
	int head;
	SlideMax():head(0){}
	void Push(int v){
		while(head<int(buf.size())){
			if(buf.back()<v)
				buf.pop_back();
			else
				break;
		}
		buf.PB(v);
	}
	int Get(){
		return buf[head];
	}
	void Pop(int v){
		if(head<int(buf.size())){
			if(buf[head]==v)
				head++;
		}
	}
};

signed main(){
	InitFact();
	int n=read();
	vi a(n);
	REP(i,n)a[i]=read()-1;
	{
		int head=-1;
		REP(i,n)
			if(a[i]!=0){
				head=i;
				break;
			}
		if(head==-1){
			print(fact[n]);
			return 0;
		}
		vi tmp;
		FOR(i,head,n)
			tmp.PB(a[i]);
		REP(i,head)
			tmp.PB(a[i]);
		a=tmp;
	}
	{
		int head=-1;
		REP(i,n)
			if(a[i]==0){
				head=i;
				break;
			}
		if(head==-1){
			Impossible();
		}
		vi tmp;
		FOR(i,head,n)
			tmp.PB(a[i]);
		REP(i,head)
			tmp.PB(a[i]);
		a=tmp;
	}
	cerr<<a<<endl;
	const pi e(-1,-1);
	vector<pi> rng(n,e);
	REP(i,n){
		if(rng[a[i]]==e){
			rng[a[i]]=pi(i,i+1);
		}else{
			rng[a[i]].second=i+1;
		}
	}
	cerr<<rng<<endl;
	{
		int len=0;
		REP(i,n)
			len+=rng[i].second-rng[i].first;
		if(len!=n){
			Impossible();
		}
	}
	int k=rng[0].second-rng[0].first;
	cerr<<k<<endl;
	if(k==n){
		print(fact[n]);
		return 0;
	}
	{
		REP(i,n){
			if(rng[i].second-rng[i].first>k){
				Impossible();
			}
		}
	}
	vi cur(n,-1);
	vi dec(n,0);
	vi fix(n,-1);
	REP(i,n)if(rng[i]!=e){
		int x=rng[i].first;
		int y=rng[i].second;
		int p=-1,q=-1;
		if(x-1>=0&&a[x-1]>i){
			for(int j=x;j<x+k&&j<n&&cur[j]==-1;j++)
				cur[j]=i;
			dec[i]=1;
			p=x;
		}
		if(y<n&&a[y]>i){
			for(int j=y-1;j>=y-k&&j>=0&&cur[j]==-1;j--)
				cur[j]=i;
			dec[i]=1;
			q=y-k;
		}
		if(p!=-1&&q!=-1&&p!=q)
			Impossible();
		if(dec[i]){
			int r=p;
			if(r==-1)r=q;
			fix[r]=i;
		}
	}
	cerr<<cur<<endl;
	cerr<<dec<<endl;
	cerr<<fix<<endl;
	{
		REP(i,n){
			if(cur[i]==-1){
				if(dec[a[i]])
					Impossible();
			}else{
				assert(dec[cur[i]]);
				if(cur[i]!=a[i])
					Impossible();
			}
		}
	}
	{
		vi pos;
		REP(i,n){
			if(fix[i]!=-1)
				pos.PB(i);
		}
		REP(i,pos.size()){
			int x=pos[i];
			int y=i+1<int(pos.size())?pos[i+1]:n;
			if(x+k<y){
				dec[a[x+k]]=2;
				FOR(j,x+1,y){
					fix[j]=a[x+k];
				}
			}
		}
	}
	cerr<<dec<<endl;
	cerr<<fix<<endl;
	vi cnt(n,0);
	{
		REP(i,n)if(fix[i]!=-1){
			cnt[fix[i]]++;
			assert(dec[fix[i]]);
		}
		SlideMax sm;
		for(int i=n-1;i>=0;i--){
			sm.Push(a[i]);
			if(i+k<n)
				sm.Pop(a[i+k]);
			if(fix[i]==-1){
				cerr<<i<<" "<<sm.Get()<<endl;
				cnt[sm.Get()]++;
			}
		}
	}
	cerr<<cnt<<endl;
	int ans=1;
	{
		int rem=0;
		REP(i,n){
			rem+=cnt[i];
			cerr<<i<<" "<<cnt[i]<<" "<<rem<<endl;
			if(dec[i]==1){
				//assert(cnt[i]==1);
				mult(ans,1);
			}else if(dec[i]==2){
				cerr<<k+1-(cnt[i]-k+1)<<endl;
				mult(ans,k+1-(cnt[i]-k+1));
			}else{
				mult(ans,rem);
			}
			rem--;
		}
	}
	print(ans);
}