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

using ll=long long;
#define int ll

#define rng(i,a,b) for(int i=int(a);i<int(b);i++)
#define rep(i,b) rng(i,0,b)
#define gnr(i,a,b) for(int i=int(b)-1;i>=int(a);i--)
#define per(i,b) gnr(i,0,b)
#define pb push_back
#define eb emplace_back
#define a first
#define b second
#define bg begin()
#define ed end()
#define all(x) x.bg,x.ed
#define si(x) int(x.size())
#ifdef LOCAL
#define dmp(x) cerr<<__LINE__<<" "<<#x<<" "<<x<<endl
#else
#define dmp(x) void(0)
#endif

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> using vc=vector<t>;
template<class t> using vvc=vc<vc<t>>;

using pi=pair<int,int>;
using vi=vc<int>;

template<class t,class u>
ostream& operator<<(ostream& os,const pair<t,u>& p){
	return os<<"{"<<p.a<<","<<p.b<<"}";
}

template<class t> ostream& operator<<(ostream& os,const vc<t>& v){
	os<<"{";
	for(auto e:v)os<<e<<",";
	return os<<"}";
}

#define mp make_pair
#define mt make_tuple
#define one(x) memset(x,-1,sizeof(x))
#define zero(x) memset(x,0,sizeof(x))
#ifdef LOCAL
void dmpr(ostream&os){os<<endl;}
template<class T,class... Args>
void dmpr(ostream&os,const T&t,const Args&... args){
	os<<t<<" ";
	dmpr(os,args...);
}
#define dmp2(...) dmpr(cerr,__LINE__,##__VA_ARGS__)
#else
#define dmp2(...) void(0)
#endif

using uint=unsigned;
using ull=unsigned long long;

template<class t,size_t n>
ostream& operator<<(ostream&os,const array<t,n>&a){
	return os<<vc<t>(all(a));
}

template<int i,class T>
void print_tuple(ostream&,const T&){
}

template<int i,class T,class H,class ...Args>
void print_tuple(ostream&os,const T&t){
	if(i)os<<",";
	os<<get<i>(t);
	print_tuple<i+1,T,Args...>(os,t);
}

template<class ...Args>
ostream& operator<<(ostream&os,const tuple<Args...>&t){
	os<<"{";
	print_tuple<0,tuple<Args...>,Args...>(os,t);
	return os<<"}";
}

template<class t>
void print(t x,int suc=1){
	cout<<x;
	if(suc==1)
		cout<<"\n";
	if(suc==2)
		cout<<" ";
}

ll read(){
	ll i;
	cin>>i;
	return i;
}

vi readvi(int n,int off=0){
	vi v(n);
	rep(i,n)v[i]=read()+off;
	return v;
}

template<class T>
void print(const vector<T>&v,int suc=1){
	rep(i,v.size())
		print(v[i],i==int(v.size())-1?suc:2);
}

string readString(){
	string s;
	cin>>s;
	return s;
}

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

//#define CAPITAL
void yes(bool ex=true){
	#ifdef CAPITAL
	cout<<"YES"<<"\n";
	#else
	cout<<"Yes"<<"\n";
	#endif
	if(ex)exit(0);
}
void no(bool ex=true){
	#ifdef CAPITAL
	cout<<"NO"<<"\n";
	#else
	cout<<"No"<<"\n";
	#endif
	if(ex)exit(0);
}
void possible(bool ex=true){
	#ifdef CAPITAL
	cout<<"POSSIBLE"<<"\n";
	#else
	cout<<"Possible"<<"\n";
	#endif
	if(ex)exit(0);
}
void impossible(bool ex=true){
	#ifdef CAPITAL
	cout<<"IMPOSSIBLE"<<"\n";
	#else
	cout<<"Impossible"<<"\n";
	#endif
	if(ex)exit(0);
}

constexpr ll ten(int n){
	return n==0?1:ten(n-1)*10;
}

const ll infLL=LLONG_MAX/3;

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

int topbit(signed t){
	return t==0?-1:31-__builtin_clz(t);
}
int topbit(ll t){
	return t==0?-1:63-__builtin_clzll(t);
}
int botbit(signed a){
	return a==0?32:__builtin_ctz(a);
}
int botbit(ll a){
	return a==0?64:__builtin_ctzll(a);
}
int popcount(signed t){
	return __builtin_popcount(t);
}
int popcount(ll t){
	return __builtin_popcountll(t);
}
bool ispow2(int i){
	return i&&(i&-i)==i;
}
ll mask(int i){
	return (ll(1)<<i)-1;
}

bool inc(int a,int b,int c){
	return a<=b&&b<=c;
}

template<class t> void mkuni(vc<t>&v){
	sort(all(v));
	v.erase(unique(all(v)),v.ed);
}

ll rand_int(ll l, ll r) { //[l, r]
	#ifdef LOCAL
	static mt19937_64 gen;
	#else
	static mt19937_64 gen(chrono::steady_clock::now().time_since_epoch().count());
	#endif
	return uniform_int_distribution<ll>(l, r)(gen);
}

template<class t>
void myshuffle(vc<t>&a){
	rep(i,si(a))swap(a[i],a[rand_int(0,i)]);
}

template<class t>
int lwb(const vc<t>&v,const t&a){
	return lower_bound(all(v),a)-v.bg;
}

//VERIFY: yosupo
template<class N>
struct segtree{
	vc<N> buf;
	int s;
	template<class t>
	segtree(vc<t> a){
		int n=a.size();
		s=1;
		while(s<n){s*=2;}
		buf.resize(s*2);
		rep(i,n)
			buf[s+i]=N(a[i]);
		gnr(i,1,s)
			buf[i]=N::merge(buf[i*2],buf[i*2+1]);
	}
	void set(int i,const N&t){
		i+=s;
		buf[i]=t;
		while(i>>=1)buf[i]=N::merge(buf[i*2],buf[i*2+1]);
	}
	N query(int b,int e,int l,int r,int i){
		if(e<=l||r<=b)return N();
		if(b<=l&&r<=e)return buf[i];
		return N::merge(query(b,e,l,(l+r)/2,i*2),query(b,e,(l+r)/2,r,i*2+1));
	}
	N query(int b,int e){
		return query(b,e,0,s,1);
	}
	N getall(){
		return buf[1];
	}
};

//simple range max
struct N{
	int w[4][4];
	N(int v=-1){
		zero(w);
		if(v>=-1)rep(i,4)w[i][i]=1;
		if(v>=0){
			//w[0][0]=1;
			w[0][1]=1;
			w[0][2]=v;
			w[0][3]=v;
			//w[1][1]=1;
			w[1][2]=v;
			w[1][3]=v;
			//w[2][2]=1;
			w[2][3]=1;
			//w[3][3]=1;
		}
	}
	void push(N&,N&){
	}
	static N merge(N x,N y){
		N res(-2);
		rep(i,4)rng(j,i,4)rng(k,i,j+1)res.w[i][j]+=x.w[i][k]*y.w[k][j];
		return res;
	}
};

signed main(){
	cin.tie(0);
	ios::sync_with_stdio(0);
	cout<<fixed<<setprecision(20);
	
	int n;cin>>n;
	vi a=readvi(n);
	vi b=readvi(n);
	
	{
		int i=max_element(all(a))-a.bg;
		rotate(a.bg,a.bg+i,a.ed);
		rotate(b.bg,b.bg+i,b.ed);
	}
	a.pb(a[0]);
	b.pb(b[0]);
	n++;
	
	//dmp(a);
	//dmp(b);
	
	segtree<N> seg(b);
	
	auto val=[&](int i,int j){
		if(i+1==j){
			return (a[i]+a[j])*(j-i);
		}
		N z=seg.query(i+1,j);
		dmp2(j-i-1,z.w[0][3]);
		return (a[i]+a[j])*(j-i)-z.w[0][3]*2;
	};
	
	int ans=0;
	
	set<int> s;
	rep(i,n)s.insert(i);
	rep(i,n-1)ans+=val(i,i+1);
	
	vi cur(n);
	priority_queue<pi> pq;
	auto upd=[&](int i){
		if(i==0||i==n-1)return;
		auto itr=s.find(i);
		assert(itr!=s.ed);
		assert(itr!=s.bg);
		assert(next(itr)!=s.ed);
		int x=*prev(itr),y=*next(itr);
		cur[i]=val(x,y)-val(x,i)-val(i,y);
		pq.emplace(cur[i],i);
	};
	
	rng(i,1,n-1)upd(i);
	
	while(si(pq)){
		int v,i;tie(v,i)=pq.top();pq.pop();
		if(cur[i]!=v)continue;
		if(cur[i]<0)break;
		ans+=cur[i];
		cur[i]=-inf;
		auto itr=s.find(i);
		assert(itr!=s.ed);
		int x=*prev(itr),y=*next(itr);
		s.erase(itr);
		upd(x);
		upd(y);
	}
	
	cout<<((long double)(ans))/(n-1)/2<<endl;
}


Measured memory_bytes: 93144