#ifndef LOCAL
#pragma GCC optimize ("Ofast")
#pragma GCC optimize ("unroll-loops")
#endif

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

using ll=long long;
#define int ll

bool dbg=false;

#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> bool chmax(t&a,u b){if(a<b){a=b;return true;}else return false;}
template<class t,class u> bool chmin(t&a,u b){if(b<a){a=b;return true;}else return false;}

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

using pi=pair<int,int>;
using vi=vc<int>;
using vvi=vc<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));
}

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);
}

ll rand_int(ll k){ //[0,k)
	return rand_int(0,k-1);
}
string rand_string(int n,char lw,char up){
	string s(n,'?');
	rep(i,n)s[i]=rand_int(lw,up);
	return s;
}

int current_run_id,run_batch_size=1000;
int calc_random_limit(){
	return current_run_id/run_batch_size+1;
}
template<class t>
void generate_single(t&a){
	a=rand_int(1,calc_random_limit());
}
void generate_single(string&a){
	int n;generate_single(n);
	a=rand_string(n,'a','b');
}
template<class t,class u>
void generate_single(pair<t,u>&a){
	generate_single(a.a);
	generate_single(a.b);
}
//https://trap.jp/post/1224/
template<class... Args>
void input(Args&... a){
	if(dbg){
		(generate_single(a),...);
	}else{
		(cin >> ... >> a);
	}
}
#define INT(...) int __VA_ARGS__;input(__VA_ARGS__)
#define LL(...) ll __VA_ARGS__;input(__VA_ARGS__)
#define ULL(...) ull __VA_ARGS__;input(__VA_ARGS__)
#define STR(...) string __VA_ARGS__;input(__VA_ARGS__)
#define CHR(...) char __VA_ARGS__;input(__VA_ARGS__)
#define DBL(...) double __VA_ARGS__;input(__VA_ARGS__)
#define LD(...) ld __VA_ARGS__;input(__VA_ARGS__)
#define overload3(a,b,c,d,...) d
#define VI2(name,size) vi name(size);rep(i_##name,size)input(name[i_##name]);
#define VI3(name,size,offset) vi name(size);rep(i_##name,size)input(name[i_##name]),name[i_##name]+=offset;
#define VI(...) overload3(__VA_ARGS__,VI3,VI2)(__VA_ARGS__)
#define VPI(name,size) vc<pi> name(size);rep(i_##name,size)input(name[i_##name]);
#define VVI(name,sizeN,sizeM) vvi name(sizeN,vi(sizeM));\
rep(i_##name,sizeN)rep(j_##name,sizeM)input(name[i_##name][j_##name]);
#define VS(name,size) vc<string> name(size);rep(i_##name,size)input(name[i_##name]);

#define overload5(a,b,c,d,e,f,...) f
#define VVC4(type,name,sizeN,sizeM) vvc<type> name(sizeN,vc<type>(sizeM));
#define VVC5(type,name,sizeN,sizeM,ini) vvc<type> name(sizeN,vc<type>(sizeM,ini));
#define VVC(...) overload5(__VA_ARGS__,VVC5,VVC4)(__VA_ARGS__)

template<class T>
T vvvc(T v){
	return v;
}

template<class T,class...Args>
auto vvvc(int n,T v,Args...args){
	return vector(n,vvvc(v,args...));
}

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<<"}";
}

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;
}

pi readpi(int off=0){
	int a,b;cin>>a>>b;
	return pi(a+off,b+off);
}

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

template<class t,class u>
void print_single(const pair<t,u>&p,int suc=1){
	print_single(p.a,2);
	print_single(p.b,suc);
}

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

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

template<class T,size_t N>
void print_single(const array<T,N>&v,int suc=1){
	rep(i,N)
		print_single(v[i],i==int(N)-1?suc:2);
}

template<class T>
void print(const T&t){
	print_single(t);
}

template<class T,class ...Args>
void print(const T&t,const Args&...args){
	print_single(t,2);
	print(args...);
}

template<class T>
void printvv(const vvc<T>&vs){
	for(const auto&row:vs)print(row);
}

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

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

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

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 topbit(ull 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 botbit(ull a){
	return a==0?64:__builtin_ctzll(a);
}
int popcount(signed t){
	return __builtin_popcount(t);
}
int popcount(ll t){
	return __builtin_popcountll(t);
}
int popcount(ull t){
	return __builtin_popcountll(t);
}
int bitparity(ll t){
	return __builtin_parityll(t);
}
bool ispow2(int i){
	return i&&(i&-i)==i;
}
ll mask(int i){
	return (ll(1)<<i)-1;
}
ull umask(int i){
	return (ull(1)<<i)-1;
}
ll minp2(ll n){
	if(n<=1)return 1;
	else return ll(1)<<(topbit(n-1)+1);
}

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

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

template<class t> bool isuni(vc<t> v){
	int s=si(v);
	mkuni(v);
	return si(v)==s;
}

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

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

//VERIFY: yosupo
//KUPC2017J
//AOJDSL1A
//without rank
struct unionfind{
	vi p,s;
	int c;
	unionfind(int n):p(n,-1),s(n,1),c(n){}
	void clear(){
		fill(all(p),-1);
		fill(all(s),1);
		c=si(p);
	}
	int find(int a){
		return p[a]==-1?a:(p[a]=find(p[a]));
	}
	//set b to a child of a
	bool unite(int a,int b){
		a=find(a);
		b=find(b);
		if(a==b)return false;
		p[b]=a;
		s[a]+=s[b];
		c--;
		return true;
	}
	bool same(int a,int b){
		return find(a)==find(b);
	}
	int sz(int a){
		return s[find(a)];
	}
};

vvc<int> readGraph(int n,int m){
	vvc<int> g(n);
	rep(i,m){
		int a,b;
		cin>>a>>b;
		//sc.read(a,b);
		a--;b--;
		g[a].pb(b);
		g[b].pb(a);
	}
	return g;
}

vvc<int> rand_tree(int n){
	vvc<int> t(n);
	unionfind uf(n);
	while(uf.c>1){
		int a=rand_int(n);
		int b=rand_int(n);
		if(uf.unite(a,b)){
			t[a].pb(b);
			t[b].pb(a);
		}
	}
	return t;
}

vvc<int> readTree(int n){
	if(dbg){
		return rand_tree(n);
	}else{
		return readGraph(n,n-1);
	}
}

void printTree(const vvc<int> t){
	int n=si(t);
	int degsum=0;
	rep(i,n)degsum+=si(t[i]);
	if(degsum==n-1){
		//directed
		rep(i,si(t))for(auto j:t[i]){
			print(i+1,j+1);
		}
	}else if(degsum==2*(n-1)){
		//undirected
		rep(i,si(t))for(auto j:t[i])if(i<j){
			print(i+1,j+1);
		}
	}else{
		assert(false);
	}
}

template<class t>
vc<t> presum(const vc<t>&a){
	vc<t> s(si(a)+1);
	rep(i,si(a))s[i+1]=s[i]+a[i];
	return s;
}
vc<ll> presum(const vi&a){
	vc<ll> s(si(a)+1);
	rep(i,si(a))s[i+1]=s[i]+a[i];
	return s;
}
//BIT で数列を管理するときに使う (CF850C)
template<class t>
vc<t> predif(vc<t> a){
	gnr(i,1,si(a))a[i]-=a[i-1];
	return a;
}
template<class t>
vvc<ll> imos(const vvc<t>&a){
	int n=si(a),m=si(a[0]);
	vvc<ll> b(n+1,vc<ll>(m+1));
	rep(i,n)rep(j,m)
		b[i+1][j+1]=b[i+1][j]+b[i][j+1]-b[i][j]+a[i][j];
	return b;
}

//verify してないや
void transvvc(int&n,int&m){
	swap(n,m);
}
template<class t,class... Args>
void transvvc(int&n,int&m,vvc<t>&a,Args&...args){
	assert(si(a)==n);
	vvc<t> b(m,vi(n));
	rep(i,n){
		assert(si(a[i])==m);
		rep(j,m)b[j][i]=a[i][j];
	}
	a.swap(b);
	transvvc(n,m,args...);
}
//CF854E
void rotvvc(int&n,int&m){
	swap(n,m);
}
template<class t,class... Args>
void rotvvc(int&n,int&m,vvc<t>&a,Args&...args){
	assert(si(a)==n);
	vvc<t> b(m,vi(n));
	rep(i,n){
		assert(si(a[i])==m);
		rep(j,m)b[m-1-j][i]=a[i][j];
	}
	a.swap(b);
	rotvvc(n,m,args...);
}

//ソートして i 番目が idx[i]
//CF850C
template<class t>
vi sortidx(const vc<t>&a){
	int n=si(a);
	vi idx(n);iota(all(idx),0);
	sort(all(idx),[&](int i,int j){return a[i]<a[j];});
	return idx;
}
//vs[i]=a[idx[i]]
//例えば sortidx で得た idx を使えば単にソート列になって返ってくる
//CF850C
template<class t>
vc<t> a_idx(const vc<t>&a,const vi&idx){
	int n=si(a);
	assert(si(idx)==n);
	vc<t> vs(n);
	rep(i,n)vs[i]=a[idx[i]];
	return vs;
}
//CF850C
vi invperm(const vi&p){
	int n=si(p);
	vi q(n);
	rep(i,n)q[p[i]]=i;
	return q;
}

template<class t,class s=t>
s SUM(const vc<t>&a){
	return accumulate(all(a),s(0));
}
template<class t,size_t K,class s=t>
s SUM(const array<t,K>&a){
	return accumulate(all(a),s(0));
}

template<class t>
t MAX(const vc<t>&a){
	return *max_element(all(a));
}

template<class t>
pair<t,int> MAXi(const vc<t>&a){
	auto itr=max_element(all(a));
	return mp(*itr,itr-a.bg);
}

template<class A>
auto MIN(const A&a){
	return *min_element(all(a));
}

template<class t>
pair<t,int> MINi(const vc<t>&a){
	auto itr=min_element(all(a));
	return mp(*itr,itr-a.bg);
}

vi vid(int n){
	vi res(n);iota(all(res),0);
	return res;
}

template<class S>
void soin(S&s){
	sort(all(s));
}

template<class S,class F>
void soin(S&s,F&&f){
	sort(all(s),forward<F>(f));
}

template<class S>
S soout(S s){
	soin(s);
	return s;
}

template<class S>
void rein(S&s){
	reverse(all(s));
}

template<class S>
S reout(S s){
	rein(s);
	return s;
}

template<class t,class u>
pair<t,u>&operator+=(pair<t,u>&a,pair<t,u> b){
	a.a+=b.a;a.b+=b.b;return a;}
template<class t,class u>
pair<t,u>&operator-=(pair<t,u>&a,pair<t,u> b){
	a.a-=b.a;a.b-=b.b;return a;}
template<class t,class u>
pair<t,u> operator+(pair<t,u> a,pair<t,u> b){return mp(a.a+b.a,a.b+b.b);}
template<class t,class u>
pair<t,u> operator-(pair<t,u> a,pair<t,u> b){return mp(a.a-b.a,a.b-b.b);}
template<class t,class u,class v>
pair<t,u>&operator*=(pair<t,u>&a,v b){
	a.a*=b;a.b*=b;return a;}
template<class t,class u,class v>
pair<t,u> operator*(pair<t,u> a,v b){return a*=b;}
template<class t,class u>
pair<t,u> operator-(pair<t,u> a){return mp(-a.a,-a.b);}
namespace std{
template<class t,class u>
istream&operator>>(istream&is,pair<t,u>&a){
	return is>>a.a>>a.b;
}
}

template<class t>
t gpp(vc<t>&vs){
	assert(si(vs));
	t res=move(vs.back());
	vs.pop_back();
	return res;
}

template<class t,class u>
void pb(vc<t>&a,const vc<u>&b){
	a.insert(a.ed,all(b));
}

template<class t,class...Args>
vc<t> cat(vc<t> a,Args&&...b){
	(pb(a,forward<Args>(b)),...);
	return a;
}

template<class t,class u>
vc<t>& operator+=(vc<t>&a,u x){
	for(auto&v:a)v+=x;
	return a;
}

template<class t,class u>
vc<t> operator+(vc<t> a,u x){
	return a+=x;
}

template<class t>
vc<t>& operator+=(vc<t>&a,const vc<t>&b){
	a.resize(max(si(a),si(b)));
	rep(i,si(b))a[i]+=b[i];
	return a;
}

template<class t>
vc<t> operator+(const vc<t>&a,const vc<t>&b){
	vc<t> c(max(si(a),si(b)));
	rep(i,si(a))c[i]+=a[i];
	rep(i,si(b))c[i]+=b[i];
	return c;
}

template<class t,class u>
vc<t>& operator-=(vc<t>&a,u x){
	for(auto&v:a)v-=x;
	return a;
}
template<class t,class u>
vc<t> operator-(vc<t> a,u x){
	return a-=x;
}

template<class t,class u>
vc<t>& operator*=(vc<t>&a,u x){
	for(auto&v:a)v*=x;
	return a;
}
template<class t,class u>
vc<t> operator*(vc<t> a,u x){
	return a*=x;
}

template<class t,class u>
vc<t>& operator/=(vc<t>&a,u x){
	for(auto&v:a)v/=x;
	return a;
}
template<class t,class u>
vc<t> operator/(vc<t> a,u x){
	return a/=x;
}

template<class t>
vc<t>& operator<<=(vc<t>&a,int k){
	assert(k>=0);
	a.insert(a.bg,k,t(0));
	return a;
}
template<class t>
vc<t> operator<<(vc<t> a,int k){
	return a<<=k;
}

template<class t,class u>
void remval(vc<t>&a,const u&v){
	a.erase(remove(all(a),v),a.ed);
}
//消した要素の個数を返してくれる
//UCUP 2-8-F
template<class t,class F>
int remif(vc<t>&a,F f){
	auto itr=remove_if(all(a),f);
	int res=a.ed-itr;
	a.erase(itr,a.ed);
	return res;
}

template<class VS,class u>
void fila(VS&vs,const u&a){
	fill(all(vs),a);
}

template<class t,class u>
int findid(const vc<t>&vs,const u&a){
	auto itr=find(all(vs),a);
	if(itr==vs.ed)return -1;
	else return itr-vs.bg;
}

template<class t>
void rtt(vc<t>&vs,int i){
	rotate(vs.bg,vs.bg+i,vs.ed);
}

//Multiuni2023-8 C
//f(lw)=false,...,f(n-1)=false,f(n)=true,...,f(up)=true,
//のときに n を返す
template<class F>
int find_min_true(int lw,int up,F f){
	while(up-lw>1){
		const int mid=(lw+up)/2;
		if(f(mid))up=mid;
		else lw=mid;
	}
	return up;
}
//f(lw)=true,f(up)=false
template<class F>
int find_max_true(int lw,int up,F f){
	while(up-lw>1){
		const int mid=(lw+up)/2;
		if(f(mid))lw=mid;
		else up=mid;
	}
	return lw;
}

template<class t> using pqmin=priority_queue<t,vc<t>,greater<t>>;
template<class t> using pqmax=priority_queue<t>;
using T=tuple<int,int,int>;

//copied from yosupo's library
//PARTLY VERIFIED

//USACO 2022 January ptlatinum C

//#define GEOF

#ifdef GEOF
using ld=long double;
//using ld=double;
const ld PI=acos(ld(-1));
#else
using ld=ll;
#endif
const ld eps=1e-9;
int sgn(ld a){return a<-eps?-1:(a>eps?1:0);}
int sgn(ld a,ld b){return sgn(a-b);}
/*
using pt=complex<ld>;
#define x real()
#define y imag()
*/
struct pt {
    ld x,y;
    //pt(ld _x = ld(0), ld _y = ld(0)) : x(_x), y(_y) {}
    pt():x(0),y(0){}
    pt(ld xx,ld yy):x(xx),y(yy){}
    pt operator+(const pt& r) const { return {x + r.x, y + r.y}; }
    pt operator-(const pt& r) const { return {x - r.x, y - r.y}; }
    pt operator*(const pt& r) const {
        return {x * r.x - y * r.y, x * r.y + y * r.x};
    }
    pt inv()const{
		ld d=norm();
		return {x/d,-y/d};
	}
    pt operator/(const pt&r)const{
		return operator*(r.inv());
	}

    pt operator*(const ld& r) const { return {x * r, y * r}; }
    pt operator/(const ld& r) const { return {x / r, y / r}; }

    pt& operator+=(const pt& r) { return *this = *this + r; }
    pt& operator-=(const pt& r) { return *this = *this - r; }
    pt& operator*=(const pt& r) { return *this = *this * r; }
    pt& operator/=(const pt& r) { return *this = *this / r; }
    pt& operator*=(const ld& r) { return *this = *this * r; }
    pt& operator/=(const ld& r) { return *this = *this / r; }

    pt operator-() const { return {-x, -y}; }

	static int cmp(const pt&a,const pt&b){
		int v=sgn(a.x,b.x);
		return v?v:sgn(a.y,b.y);
	}
    bool operator<(const pt& r) const {
        return cmp(*this,r)<0;
    }
    bool operator<=(const pt& r) const {
        return cmp(*this,r)<=0;
    }
    bool operator>(const pt& r) const {
        return cmp(*this,r)>0;
    }
    bool operator==(const pt& r) const { return sgn((*this - r).rabs()) == 0; }
    bool operator!=(const pt& r) const { return !(*this == r); }

    ld norm() const { return x * x + y * y; }
    ld rabs() const { return max(std::abs(x), std::abs(y)); }  // robust abs
    pair<ld, ld> to_pair() const { return {x, y}; }
    #ifdef GEOF
    ld abs() const { return sqrt(norm()); }
    ld arg() const { return atan2(y, x); }
    static pt polar(ld le, ld th) { return {le * cos(th), le * sin(th)}; }
	#endif
};
istream& operator>>(istream& is, pt& p){
	return is>>p.x>>p.y;
}
ostream& operator<<(ostream& os, const pt& p) {
    return os << "pt(" << p.x << ", " << p.y << ")";
}
ld norm(const pt&a){
	return a.norm();
}
ld rabs(const pt&a){
	return a.rabs();
}
#ifdef GEOF
ld abs(const pt&a){
	return sqrt(norm(a));
}
//XXII Opencup Gpt of Ural K
pt normalized(const pt&a){
	return a/abs(a);
}
ld arg(const pt&a){return a.arg();}
//normalize to [-PI,PI)
//Contest 2: ptKU Contest 1, ptTZ Summer 2022 Day 4
ld normarg(ld a){
	ld res=fmod(a+PI,2*PI);
	if(res<0)res+=PI;
	else res-=PI;
	return res;
}
//normalize to [0,2*PI)
//Multiuni2023-10-E
ld normarg_nonnegative(ld a){
	ld res=fmod(a,2*PI);
	if(res<0)res+=2*PI;
	return res;
}
//AOJ1183
//arg between ab
//assume given lengths are valid
ld arg(ld a,ld b,ld c){
	return acos(min(max((a*a+b*b-c*c)/(2*a*b),ld(-1)),ld(1)));
}
//UCUP 2-20-D
ld heron(ld a,ld b,ld c){
	ld s=(a+b+c)/2;
	return sqrt(s*(s-a)*(s-b)*(s-c));
}
#endif
ld norm(const pt&a,const pt&b){
	return (a-b).norm();
}
ld dot(const pt&a,const pt&b){return a.x*b.x+a.y*b.y;}
ld crs(const pt& a,const pt& b){return a.x*b.y-a.y*b.x;}
ld crs(const pt& a,const pt& b,const pt& c){return crs(b-a,c-a);}
int ccw(const pt& a,const pt& b){return sgn(crs(a,b));}
int ccw(const pt& a,const pt& b,const pt& c){return ccw(b-a,c-a);}
//(-pi,0](0,pi]
int argtype(const pt&a){
	if(sgn(a.y)==0)return a.x<0?1:0;
	return a.y<0?0:1;
}
int argcmp(const pt&a,const pt&b){
	int at=argtype(a),bt=argtype(b);
	if(at!=bt)return at<bt?-1:1;
	return -ccw(a,b);
};
bool argless(const pt&a,const pt&b){return argcmp(a,b)<0;}
//c の位置を聞く関数です，b じゃないです
//(-2)[a,-1](0)[b,1](2)
int bet(pt a,pt b,pt c){
	pt d=b-a;
	ld e=dot(d,c-a);
	if(sgn(e)<=0)return sgn(e)-1;
	return sgn(e-norm(d))+1;
}
//AOJ0153
//三角形 abc に d が含まれるか？0-no,1-edge,2-in
int cont(pt a,pt b,pt c,pt d){
	if(ccw(a,b,c)==-1) swap(b,c);
	return min({ccw(a,b,d),ccw(b,c,d),ccw(c,a,d)})+1;
}
//(a,b) を結ぶ直線を考え，x 座標との交点の y 座標を求める
//(分子,分母)を返す
pair<ld,ld> xcut(const pt&a,const pt&b,ld x){
	return mp(a.y*(b.x-x)-b.y*(a.x-x),b.x-a.x);
}
//XXII Opencup Gpt of Ural K
pt rot90(pt a){
	return pt(-a.y,a.x);
}
#ifdef GEOF
//Multiuni 2024-6-C
pt rot(pt a,ld b){
	ld c=cos(b),s=sin(b);
	return pt(a.x*c-a.y*s,a.x*s+a.y*c);
}
ld xcutval(const pt&a,const pt&b,ld x){
	auto [p,q]=xcut(a,b,x);
	return p/q;
}
//AOJ1183
//Xmas2010 E
//-+ の 順で返す
//a の符号により，small/large が決まる
int qeq(ld a,ld b,ld c,ld&d,ld&e){
	if(sgn(a)==0){
		if(sgn(b)==0)return 0;
		d=-c/b;
		return 1;
	}
	ld f=b*b-4*a*c;
	if(sgn(f)<0)return 0;
	ld g=sqrt(max(f,ld(0)));
	d=(-b-g)/(2*a);
	e=(-b+g)/(2*a);
	return sgn(f)+1;
}
#else
pt normdir(pt a){
	if(a==pt(0,0))return a;
	int g=gcd(a.x,a.y);
	return pt(a.x/g,a.y/g);
}
#endif

ld area2(const vc<pt>&ps){
	ld res=0;
	rep(i,si(ps))res+=crs(ps[i],ps[(i+1)%si(ps)]);
	return res;
}

//stress-test してません
vc<pt> lower_convex(const vc<pt>&ps,bool onedge){
	vc<pt> res;
	assert(is_sorted(all(ps)));
	for(auto p:ps){
		int s;
		while((s=si(res))>=2){
			if(ccw(res[s-2],res[s-1],p)<=(onedge?-1:0))res.pop_back();
			else break;
		}
		res.pb(p);
	}
	return res;
}
vc<pt> upper_convex(const vc<pt>&ps,bool onedge){
	vc<pt> res;
	assert(is_sorted(all(ps)));
	for(auto p:ps){
		int s;
		while((s=si(res))>=2){
			if(ccw(res[s-2],res[s-1],p)>=(onedge?1:0))res.pop_back();
			else break;
		}
		res.pb(p);
	}
	return res;
}
vc<pt> convex(vc<pt> ps,bool onedge){
	mkuni(ps);
	auto lw=lower_convex(ps,onedge);
	auto up=upper_convex(ps,onedge);
	lw.pop_back();
	rein(up);
	up.pop_back();
	pb(lw,up);
	return lw;
}

//GCJ2019R3D
//0-out,1-edge,2-in
int cont(const vc<pt>&a,pt b){
	int n=a.size();
	int res=0;
	rep(i,n){
		pt c=a[i],d=a[(i+1)%n];
		if(ccw(c,d,b)==0&&abs(bet(c,d,b))<=1)
			return 1;
		if(c.y>d.y)swap(c,d);
		if(sgn(b.y-c.y)>0&&sgn(d.y-b.y)>=0&&ccw(c,d,b)==1)
			res^=1;
	}
	return res*2;
}

//長さ N,要素 0-M-1 のソート列を全探索し，F を呼ぶ
template<class F>
void sorted_array_rec(int n,int m,F f){
	vi a(n);
	auto rec=[&](auto self,int i)->void{
		if(i==n)f(a);
		else{
			int last=i?a[i-1]:0;
			rng(nx,last,m){
				a[i]=nx;
				self(self,i+1);
			}
		}
	};
	rec(rec,0);
}

//UCUP 2-27-B
//[0,m) の値を distinct に n 個取ってきたやつを並べる
template<class F>
void sorted_distinct_rec(int n,int m,F f){
	sorted_array_rec(n,m-n+1,[&](vi a){
		rep(i,n)a[i]+=i;
		f(a);
	});
}

//vs から k 要素選んだ tuple すべてに対して f を呼ぶ
//ARC177 E
template<class t,class F>
void tuple_rec(const vc<t>&vs,int k,F f){
	int n=si(vs);
	if(k<=n){
		vc<t> a(k);
		sorted_array_rec(k,n-k+1,[&](const vi&idx){
			rep(i,k)a[i]=vs[idx[i]+i];
			f(a);
		});
	}
};

template<class F> void permrep(int n,F f){
	vi p(n);iota(all(p),0);
	do{
		f(p);
	}while(next_permutation(all(p)));
};

//N() が単位元
//VERIFY: yosupo
//CF407E
template<class N>
struct segtree{
	vc<N> x;
	int n,s;
	segtree(){}
	template<class t>
	segtree(const vc<t>&a){
		n=a.size();
		s=1;
		while(s<n){s*=2;}
		x.resize(s*2);
		rep(i,n)
			x[s+i]=N(a[i]);
		gnr(i,1,s)
			x[i]=N::merge(x[i*2],x[i*2+1]);
	}
	//NOT Verified
	segtree(int nn){
		resize(nn);
	}
	void resize(int nn){
		n=nn;
		s=1;
		while(s<n){s*=2;}
		x.assign(s*2,N());
		gnr(i,1,s)
			x[i]=N::merge(x[i*2],x[i*2+1]);
	}
	template<class t>
	void init(const vc<t>&a){
		n=a.size();
		s=1;
		while(s<n){s*=2;}
		x.resize(s*2);
		rep(i,n)
			x[s+i]=N(a[i]);
		rng(i,n,s)
			x[s+i]=N();
		gnr(i,1,s)
			x[i]=N::merge(x[i*2],x[i*2+1]);
	}
	void clear(){
		rep(i,n)
			x[s+i]=N();
		gnr(i,1,s)
			x[i]=N::merge(x[i*2],x[i*2+1]);
	}
	N point_get(int i){
		assert(inc(0,i,n-1));
		return x[i+s];
	}
	void point_set(int i,const N&t){
		assert(inc(0,i,n-1));
		i+=s;
		x[i]=t;
		while(i>>=1)x[i]=N::merge(x[i*2],x[i*2+1]);
	}
	void point_merge(int i,const N&t){
		assert(inc(0,i,n-1));
		i+=s;
		x[i]=N::merge(x[i],t);
		while(i>>=1)x[i]=N::merge(x[i*2],x[i*2+1]);
	}
	template<class F,class...Args>
	void point_change(int i,F f,Args&&...args){
		assert(inc(0,i,n-1));
		i+=s;
		(x[i].*f)(forward<Args>(args)...);
		while(i>>=1)x[i]=N::merge(x[i*2],x[i*2+1]);
	}
	N composite(int b,int e){
		assert(0<=b&&b<=e&&e<=n);
		N lf,rt;
		for(int l=b+s,r=e+s;l<r;l>>=1,r>>=1){
			if (l&1){
				lf=N::merge(lf,x[l]);
				l++;
			}
			if (r&1){
				r--;
				rt=N::merge(x[r],rt);
			}
		}
		return N::merge(lf,rt);
	}
	N getall(){
		return x[1];
	}
	//UTPC2020E
	//n 超えるかもしれない
	template <class F,class... Args> 
	pair<int,N> max_right(int l,F f,Args&&... args){
		assert((N().*f)(forward<Args>(args)...));
		assert(0<=l&&l<=n);
		if(l==n)return mp(n,N());
		l+=s;
		
		N sm;
		assert((sm.*f)(forward<Args>(args)...));
		do {
			while (l % 2 == 0) l >>= 1;
			if (!(N::merge(sm,x[l]).*f)(forward<Args>(args)...)){
				while (l < s) {
					l = (2 * l);
					N tmp=N::merge(sm,x[l]);
					if ((tmp.*f)(forward<Args>(args)...)) {
						sm = tmp;
						l++;
					}
				}
				return mp(l - s,sm);
			}
			sm = N::merge(sm, x[l]);
			l++;
		} while ((l & -l) != l);
		return mp(n,sm);
	}
	//UCUP-2-22-K
	template <class F,class... Args> 
	pair<int,N> min_left_withinit(int r,N sm,F f,Args&&... args){
		assert((sm.*f)(forward<Args>(args)...));
		assert(0<=r&&r<=n);
        if(r==0)return mp(0,sm);
        r+=s;
        do {
            r--;
            while (r > 1 && (r % 2)) r >>= 1;
            if (!(N::merge(x[r],sm).*f)(forward<Args>(args)...)) {
                while (r < s) {
                    r = (2 * r + 1);
                    N tmp=N::merge(x[r],sm);
                    if ((tmp.*f)(forward<Args>(args)...)) {
                        sm = tmp;
                        r--;
                    }
                }
                return mp(r + 1 - s,sm);
            }
            sm = N::merge(x[r], sm);
        } while ((r & -r) != r);
        return mp(0,sm);
    }
	//UTPC2020E
	template <class F,class... Args> 
	pair<int,N> min_left(int r,F f,Args&&... args){
		return min_left_withinit(r,N(),f,forward<Args>(args)...);
    }
    //行列とか乗せて必要なのはベクトルとの積，みたいなときに使える？
    //CF Goodbye 2016 E
    //CF 896 F
	template<class F,class T,class... Args>
	T accumulate(int l,int r,F f,T t,Args&&... args) {
		assert(0<=l&&l<=r&&r<=n);
		static int buf[2][30];
		int cnt[2]{};
		for(l+=s,r+=s;l<r;l>>=1,r>>=1){
			if(l&1)buf[0][cnt[0]++]=l;
			if(r&1)buf[1][cnt[1]++]=r-1;
			l++;
		}
		rep(i,cnt[0])t=(x[buf[0][i]].*f)(t,forward<Args>(args)...);
		per(i,cnt[1])t=(x[buf[1][i]].*f)(t,forward<Args>(args)...);
		return t;
	}
};

struct MinNode{
	pi v;
	MinNode(pi vv=pi(inf,-1)):v(vv){}
	MinNode(int val,int idx):v(val,idx){}
	static MinNode merge(const MinNode&a,const MinNode&b){
		return MinNode(min(a.v,b.v));
	}
};

bool brutecheck(vc<pt> ps){
	assert(isuni(ps));
	int n=si(ps);assert(n%2==0);
	int pre=1;
	rep(i,n+1){
		pt a=ps[(i+0)%n];
		pt b=ps[(i+1)%n];
		pt c=ps[(i+2)%n];
		int k=ccw(a,b,c);
		assert(k!=0);
		if(pre==k)return false;
		pre=k;
	}
	return true;
}

void slv(){
	INT(n);
	vc<pt> ps(n);
	rep(i,n)cin>>ps[i];
	auto cv=convex(ps,false);
	int inner=n-si(cv);
	if(inner<=1){
		return print(-1);
	}else if(inner==2){
		pt A,B;
		{
			vc<pt> in;
			auto vs=cv;
			soin(vs);
			for(auto p:ps)if(!bis(vs,p)){
				in.pb(p);
			}
			A=in[0];
			B=in[1];
		}
		vc<pt> lf,rt;
		ps=cv;
		n=si(cv);
		rep(i,n){
			if(ccw(A,B,ps[i])==-1){
				rtt(ps,i);
				break;
			}
		}
		rep(i,n){
			if(ccw(A,B,ps[i])==1){
				rtt(ps,i);
				break;
			}
		}
		for(auto p:ps){
			if(ccw(A,B,p)==1){
				lf.pb(p);
			}else{
				rt.pb(p);
			}
		}
		
		const int V=si(rt);
		segtree<MinNode> seg(V+1);
		
		vc<pi> ab;
		
		//vc<R> ue;
		rep(i,si(lf)){
			auto p=lf[i];
			int a=find_min_true(0,V,[&](int x){
				return ccw(p,A,rt[x])<0;
			});
			int b=find_min_true(0,V,[&](int x){
				return ccw(p,B,rt[x])<0;
			});
			//ue [a,b)*[b,V)
			//sita [0,a),[a,b)
			ab.eb(a,b);
		}
		
		pi nice(-1,-1);
		rep(i,si(lf)){
			auto [a,b]=ab[i];
			if(0<a&&a<b){
				auto [v,pos]=seg.composite(0,a).v;
				if(v<b){
					nice=pi(i,pos);
				}
			}
			if(a<b&&b<V){
				seg.point_merge(a,pi(b,i));
			}
		}
		
		if(nice==pi(-1,-1)){
			return print(-1);
		}
		
		ps=vc<pt>{A,B};
		ps.pb(lf[nice.a]);
		ps.pb(lf[nice.b]);
		{
			int x=ab[nice.b].a;
			int y=ab[nice.a].b-1;
			ps.pb(rt[x]);
			ps.pb(rt[y]);
		}
	}else{
		vc<pt> in,out;
		{
			auto vs=cv;
			soin(vs);
			for(auto p:ps)if(!bis(vs,p)){
				in.pb(p);
			}
		}
		assert(si(in)>=3);
		in.resize(3);
		in=convex(in,false);
		assert(si(in)==3);
		auto kind=[&](pt p){
			int res=0;
			rep(k,3){
				pt A=in[k],B=in[(k+1)%3];
				if(ccw(A,B,p)<0)res|=1<<k;
			}
			return res;
		};
		dmp(cv);
		rep(i,si(cv))dmp(kind(cv[i]));
		rep(k,3){
			int tar=mask(3)-(1<<k);
			dmp(tar);
			bool done=false;
			for(auto p:cv)if((kind(p)&tar)==tar){
				out.pb(p);
				done=true;
				break;
			}
			if(!done){
				pt x=in[k],y=in[(k+1)%3],z=in[(k+2)%3];
				rep(i,si(cv)){
					pt p=cv[i],q=cv[(i+1)%si(cv)];
					if(ccw(x,z,p)<0&&ccw(y,z,q)>0){
						out.pb(p);
						out.pb(q);
						done=true;
						break;
					}
				}
			}
			assert(done);
			dmp(out);
		}
		mkuni(out);
		assert(si(out)<=6);
		out=convex(out,false);
		dmp(in);
		dmp(out);
		
		for(auto p:in)assert(cont(out,p)==2);
		
		ps=cat(in,out);
	}
	assert(inc(6,si(ps),9));
	
	vc<pt> ans;
	tuple_rec(ps,6,[&](vc<pt> ls){
		permrep(6,[&](vi p){
			auto now=a_idx(ls,p);
			if(brutecheck(now)){
				ans=now;
			}
		});
	});
	assert(si(ans)==6);
	print(6);
	for(auto [x,y]:ans){
		print(x,y);
	}
}

signed main(signed argc,char*argv[]){
	if(argc>1&&strcmp(argv[1],"D")==0)dbg=true;
	
	cin.tie(0);
	ios::sync_with_stdio(0);
	cout<<fixed<<setprecision(20);
	
	if(dbg){
		while(1){
			if(current_run_id%run_batch_size==0){
				cerr<<"Current Run "<<current_run_id<<endl;
			}
			slv();
			current_run_id++;
		}
	}else{
		//int t;cin>>t;rep(_,t)
		slv();
	}
}