#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,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
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;
}

using ln=pair<pt,pt>;
pt dir(ln a){return a.b-a.a;}
pt eval(ln a,ld b){return a.a+dir(a)*b;}
ld crs(ln a,pt b){return crs(a.a,a.b,b);}
int ccw(ln a,pt b){return ccw(a.a,a.b,b);}
int bet(ln a,pt b){return bet(a.a,a.b,b);}
ld projt(ln a,pt b){
	pt c=dir(a);
	return dot(c,b-a.a)/norm(c);
}
pt proj(ln a,pt b){
	pt c=dir(a);
	return a.a+c*dot(c,b-a.a)/norm(c);
}
pt refl(ln a,pt b){
	return proj(a,b)*2-b;
}
//AOJ1157
//0-no,1-yes(endpoint),2-yes(innner),3-overelap
//if the two line touch like this
// x----x----x
//it returns 1
int iss(ln a,ln b){
	int c1=ccw(a.a,a.b,b.a),c2=ccw(a.a,a.b,b.b);
	int d1=ccw(b.a,b.b,a.a),d2=ccw(b.a,b.b,a.b);
	if(c1||c2||d1||d2)return 1-max(c1*c2,d1*d2);
	int f=bet(a.a,a.b,b.a),g=bet(a.a,a.b,b.b);
	if(max(f,g)==-2||min(f,g)==2)return 0;
	if(max(f,g)==-1||min(f,g)==1)return 1;
	return 3;
}
//segment a intersects line b?
//endpoints inclusive
bool isl(ln a,ln b){
	int d1=ccw(b.a,b.b,a.a),d2=ccw(b.a,b.b,a.b);
	return d1*d2<=0;
}
//並行でない->true, というだけ
//直線が一致とかは考えてないことに注意
bool ill(ln a,ln b){
	return ccw(dir(a),dir(b));
}
ld cllt(ln a,ln b){
	return crs(b.a,b.b,a.a)/crs(dir(a),dir(b));
}
//ICPC Yokohama 2022 J
pair<ld,ld> clltf(ln a,ln b){
	return mp(crs(b.a,b.b,a.a),crs(dir(a),dir(b)));
}
//AOJ1033
pt cll(ln a,ln b){
	return eval(a,crs(b.a,b.b,a.a)/crs(dir(a),dir(b)));
}
//UCUP3-4-H
bool isp(ln a,pt b){
	return ccw(a,b)==0&&inc(-1,bet(a.a,a.b,b),1);
}
#ifdef GEOF
//AOJ2201
ld dlp(ln a,pt b){
	return abs(crs(a,b)/abs(dir(a)));
}
//AOJ0153
ld dsp(ln a,pt b){
	pt c=proj(a,b);
	if(abs(bet(a.a,a.b,c))<=1)return abs(b-c);
	return min(abs(b-a.a),abs(b-a.b));
}
//ABC314H
//b から最も近い a 上の点を返す
pt dsp_tar(ln a,pt b){
	pt c=proj(a,b);
	if(abs(bet(a.a,a.b,c))<=1)return c;
	return abs(b-a.a)<abs(b-a.b)?a.a:a.b;
}
//AOJ1157
ld dss(ln a,ln b){
	if(iss(a,b))return 0;
	return min({dsp(a,b.a),dsp(a,b.b),dsp(b,a.a),dsp(b,a.b)});
}
//AOJ2160
//反時計回り方向に伸びる垂直二等分線
ln vbis(pt a,pt b){
	pt c=(a+b)*ld(0.5),d=b-a;
	return ln(c,pt(c.x-d.y,c.y+d.x));
}
ld cutareat(ln z,ld l,ld r){
	pt a=eval(z,l);
	pt b=eval(z,r);
	return -(b.x-a.x)*(a.y+b.y)/2;
}
#endif
//ABC286H
//simple polygon と線分が交わるか
//接している場合も true
/*
bool icl(const vc<pt>&ps,ln z){
	int n=si(ps);
	rep(i,n){
		pt p=ps[i],q=ps[(i+1)%n];
		if(iss(z,ln(p,q)))return true;
	}
	return cont(ps,z.a);
}*/


//線分が polygon の中に含まれているかどうか
//polygon の周に乗っている場合も OK
//ICPC Yokohama 2022 J
bool within_polygon(const vc<pt>&ps,ln z){
	int n=si(ps);
	bool ok=true;
	int cnt=0;
	auto add=[&](ld num,ld den){
		if(den<0){
			num=-num;
			den=-den;
		}
		if(num<=0){
			cnt++;
		}else if(num>=den){
		}else{
			ok=false;
		}
	};
	rep(i,n){
		pt p=ps[i],q=ps[(i+1)%n],r=ps[(i+n-1)%n];
		int a=ccw(z,p),b=ccw(z,q);
		if(a*b==-1){
			auto [num,den]=clltf(z,ln(p,q));
			add(num,den);
		}
		if(a==0){
			int c=ccw(z,r);
			if(b!=c&&(b*c<0||ccw(p,q,r)>0))
				add(dot(ps[i]-z.a,dir(z)),norm(dir(z)));
		}
	}
	return ok&&cnt%2==1;
}

bool is_simple_polygon(const vc<pt>&ps){
	int n=si(ps);
	rep(i,n)rng(j,i+1,n){
		ln a(ps[i],ps[i+1]);
		ln b(ps[j],ps[(j+1)%n]);
		int z=iss(a,b);
		//0-no,1-yes(endpoint),2-yes(innner),3-overelap
		//if the two line touch like this
		// x----x----x
		//it returns 1
		
		if(z==0){
			//ok
		}else if(z==1){
			if(j==i+1||(i==0&&j==n-1)){
				//ok
			}else{
				return false;
			}
		}else{
			return false;
		}
	}
	return true;
}

vc<pt> generete_simple_polygon(int n){
	while(1){
		const int vmax=ten(6);
		set<pi>xy;
		while(si(xy)<n){
			int x=rand_int(-vmax,vmax);
			int y=rand_int(-vmax,vmax);
			xy.emplace(x,y);
		}
		vc<pt> ps;
		for(auto [x,y]:xy)
			ps.eb(x,y);
		myshuffle(ps);
		
		if(is_simple_polygon(ps)){
			int a=area2(ps);
			if(a<0){
				rein(ps);
			}
			return ps;
		}
	}
}


//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;
}

bool intersect_polygon(const vc<pt>&a,const vc<pt>&b){
	for(auto v:a)if(cont(b,v))return true;
	for(auto v:b)if(cont(a,v))return true;
	rep(i,si(a)){
		ln c(a[i],a[(i+1)%si(a)]);
		rep(j,si(b)){
			ln d(b[j],b[(j+1)%si(b)]);
			if(iss(c,d))return true;
		}
	}
	return false;
}


template<class F>
void subset_for(int b,F f){
	//subset of b(descending)
	int a=b;
	do{
		f(a);
		a=(a-1)&b;
	}while(a!=b);
}

void slv(){
	INT(n);
	if(dbg){
		n+=2;
	}
	vc<pt> ps(n);
	if(dbg){
		ps=generete_simple_polygon(n);
		assert(is_simple_polygon(ps));
	//int n=si(ps);
	/*rep(i,n)rng(j,i+1,n){
		ln a(ps[i],ps[i+1]);
		ln b(ps[j],ps[(j+1)%n]);
		int z=iss(a,b);
		//0-no,1-yes(endpoint),2-yes(innner),3-overelap
		//if the two line touch like this
		// x----x----x
		//it returns 1
		dmp2(a,b,z);
		
		if(z==0){
			//ok
		}else if(z==1){
			if(j==i+1||(i==0&&j==n-1)){
				//ok
			}else{
				//return false;
			}
		}else{
			//return false;
		}
	}
	//return true;*/
	}else{
		rep(i,n)cin>>ps[i];
	}
	INT(W,C);
	
	vc<pi> es;
	//VVC(bool,avail,n,n);
	rep(i,n)rng(j,i+1,n){
		if(within_polygon(ps,ln(ps[i],ps[j]))){
			bool bad=false;
			rep(k,n)if(i!=k&&j!=k)
				if(isp(ln(ps[i],ps[j]),ps[k])){
					bad=true;
					break;
				}
			if(!bad){
				//avail[i][j]=true;
				//avail[j][i]=true;
				es.eb(i,j);
				es.eb(j,i);
			}
		}
	}
	//dmp(es);
	
	soin(es,[&](pi a,pi b){
		pt c=ps[a.b]-ps[a.a];
		pt d=ps[b.b]-ps[b.a];
		int z=argcmp(c,d);
		if(z)return z<0;
		pt w=normdir(c);
		return dot(w,ps[a.a])<dot(w,ps[b.a]);
	});
	VVC(int,idx,n,n,-1);
	{
		vc<pi> tmp;
		for(auto [i,j]:es){
			if(i<j)tmp.eb(i,j);
			else idx[i][j]=si(tmp);
		}
		es.swap(tmp);
	}
	
	VVC(int,dp,n,n,0);
	vi buf(n),has(n);
	
	per(l,n)rng(r,l+1,n){
		if(int ini=idx[r][l];ini!=-1){
			fila(buf,-inf);
			fila(has,-inf);
			buf[l]=crs(ps[r],ps[l])+W+C;
			auto adv=[&](int u,int v){
				assert(u<v);
				if(l<=u&&v<=r){
					{
						int val=buf[u]+crs(ps[u],ps[v])+W+dp[u+1][v-1];
						if(ccw(ps[l],ps[r],ps[v]))
							chmax(has[v],val);
						else
							chmax(buf[v],val);
					}
					{
						int val=has[u]+crs(ps[u],ps[v])+W+dp[u+1][v-1];
						chmax(has[v],val);
					}
				}
			};
			rng(i,ini,si(es))adv(es[i].a,es[i].b);
			rep(i,ini)adv(es[i].a,es[i].b);
			chmax(dp[l][r],has[r]);
		}
		//chmax(dp[l][r],dp[l+1][r]);
		//chmax(dp[l][r],dp[l][r-1]);
		rng(mid,l,r){
			chmax(dp[l][r],dp[l][mid]+dp[mid+1][r]);
		}
	}
	
	int ans=dp[0][n-1];
	
	if(dbg){
		vi city(1<<n);
		vvc<pt> unko(1<<n);
		rep(bit,1<<n){
			vc<pt> ls;
			rep(i,n)if(bit&1<<i){
				ls.pb(ps[i]);
			}
			int k=si(ls);
			if(k>=3){
				bool ok=true;
				rep(i,k){
					int j=(i+1)%k;
					if(!within_polygon(ps,ln(ls[i],ls[j]))){
						ok=false;
						break;
					}
				}
				if(ok){
					rep(i,k){
						pt a=ls[i];
						pt b=ls[(i+1)%k];
						pt c=ls[(i+2)%k];
						if(ccw(a,b,c)<0)ok=false;
					}
				}
				if(area2(ls)&&ok){
					city[bit]=area2(ls)+W*k+C;
					unko[bit]=ls;
				}
			}
		}
		
		int god=0;
		vi st;
		auto dfs=[&](auto self,int bit)->void{
			if(bit==0){
				int sum=0;
				for(auto i:st)sum+=city[i];
				chmax(god,sum);
			}else{
				int q=botbit(bit);
				{
					self(self,bit^(1<<q));
				}
				subset_for(bit^(1<<q),[&](int use){
					use|=1<<q;
					if(si(unko[use])){
						bool ok=true;
						for(auto pre:st){
							if(intersect_polygon(unko[pre],unko[use])){
								ok=false;
							}
						}
						if(ok){
							st.pb(use);
							self(self,bit^use);
							st.pop_back();
						}
					}
				});
			}
		};
		dfs(dfs,mask(n));
		
		if(god!=ans){
			cerr<<ps<<endl;
			cerr<<W<<" "<<C<<endl;
			cerr<<god<<" "<<ans<<endl;
		}
		assert(god==ans);
	}else{
		print(ans);
	}
}

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();
	}
}