#include "squad.h" // nice trick ...
#include <bits/stdc++.h>
#include <ext/pb_ds/tree_policy.hpp>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/rope>
using namespace std;
using namespace __gnu_pbds;
using namespace __gnu_cxx;
typedef long long ll;
typedef long double ld;
typedef complex<ld> cd;
typedef pair<int, int> pi;
typedef pair<ll,ll> pl;
typedef pair<ld,ld> pd;
typedef vector<int> vi;
typedef vector<ld> vd;
typedef vector<ll> vl;
typedef vector<pi> vpi;
typedef vector<pl> vpl;
typedef vector<cd> vcd;
template <class T> using Tree = tree<T, null_type, less<T>, rb_tree_tag,tree_order_statistics_node_update>;
#define FOR(i, a, b) for (int i = (a); i < (b); i++)
#define F0R(i, a) for (int i = 0; i < (a); i++)
#define FORd(i,a,b) for (int i = (b)-1; i >= (a); i--)
#define F0Rd(i,a) for (int i = (a)-1; i >= 0; i--)
#define trav(a, x) for (auto& a : x)
#define mp make_pair
#define pb push_back
#define f first
#define s second
#define lb lower_bound
#define ub upper_bound
#define sz(x) (int)x.size()
#define all(x) begin(x), end(x)
#define rsz resize
const int MOD = 1000000007; // 998244353
const ll INF = 1e18;
const int MX = 200005;
const ld PI = 4*atan((ld)1);
template<class T> void ckmin(T &a, T b) { a = min(a, b); }
template<class T> void ckmax(T &a, T b) { a = max(a, b); }
namespace input {
template<class T> void re(complex<T>& x);
template<class T1, class T2> void re(pair<T1,T2>& p);
template<class T> void re(vector<T>& a);
template<class T, size_t SZ> void re(array<T,SZ>& a);
template<class T> void re(T& x) { cin >> x; }
void re(double& x) { string t; re(t); x = stod(t); }
void re(ld& x) { string t; re(t); x = stold(t); }
template<class Arg, class... Args> void re(Arg& first, Args&... rest) {
re(first); re(rest...);
}
template<class T> void re(complex<T>& x) { T a,b; re(a,b); x = cd(a,b); }
template<class T1, class T2> void re(pair<T1,T2>& p) { re(p.f,p.s); }
template<class T> void re(vector<T>& a) { F0R(i,sz(a)) re(a[i]); }
template<class T, size_t SZ> void re(array<T,SZ>& a) { F0R(i,SZ) re(a[i]); }
}
using namespace input;
namespace output {
template<class T1, class T2> void pr(const pair<T1,T2>& x);
template<class T, size_t SZ> void pr(const array<T,SZ>& x);
template<class T> void pr(const vector<T>& x);
template<class T> void pr(const set<T>& x);
template<class T1, class T2> void pr(const map<T1,T2>& x);
template<class T> void pr(const T& x) { cout << x; }
template<class Arg, class... Args> void pr(const Arg& first, const Args&... rest) {
pr(first); pr(rest...);
}
template<class T1, class T2> void pr(const pair<T1,T2>& x) {
pr("{",x.f,", ",x.s,"}");
}
template<class T> void prContain(const T& x) {
pr("{");
bool fst = 1; for (const auto& a: x) pr(!fst?", ":"",a), fst = 0; // const needed for vector<bool>
pr("}");
}
template<class T, size_t SZ> void pr(const array<T,SZ>& x) { prContain(x); }
template<class T> void pr(const vector<T>& x) { prContain(x); }
template<class T> void pr(const set<T>& x) { prContain(x); }
template<class T1, class T2> void pr(const map<T1,T2>& x) { prContain(x); }
void ps() { pr("\n"); }
template<class Arg> void ps(const Arg& first) {
pr(first); ps(); // no space at end of line
}
template<class Arg, class... Args> void ps(const Arg& first, const Args&... rest) {
pr(first," "); ps(rest...); // print w/ spaces
}
}
using namespace output;
namespace io {
void setIn(string s) { freopen(s.c_str(),"r",stdin); }
void setOut(string s) { freopen(s.c_str(),"w",stdout); }
void setIO(string s = "") {
ios_base::sync_with_stdio(0); cin.tie(0); // fast I/O
if (sz(s)) { setIn(s+".in"), setOut(s+".out"); } // for USACO
}
}
using namespace io;
template<class T> T invGeneral(T a, T b) {
a %= b; if (a == 0) return b == 1 ? 0 : -1;
T x = invGeneral(b,a);
return x == -1 ? -1 : ((1-(ll)b*x)/a+b)%b;
}
template<class T> struct modular {
T val;
explicit operator T() const { return val; }
modular() { val = 0; }
modular(const ll& v) {
val = (-MOD <= v && v <= MOD) ? v : v % MOD;
if (val < 0) val += MOD;
}
// friend ostream& operator<<(ostream& os, const modular& a) { return os << a.val; }
friend void pr(const modular& a) { pr(a.val); }
friend void re(modular& a) { ll x; re(x); a = modular(x); }
friend bool operator==(const modular& a, const modular& b) { return a.val == b.val; }
friend bool operator!=(const modular& a, const modular& b) { return !(a == b); }
friend bool operator<(const modular& a, const modular& b) { return a.val < b.val; }
modular operator-() const { return modular(-val); }
modular& operator+=(const modular& m) { if ((val += m.val) >= MOD) val -= MOD; return *this; }
modular& operator-=(const modular& m) { if ((val -= m.val) < 0) val += MOD; return *this; }
modular& operator*=(const modular& m) { val = (ll)val*m.val%MOD; return *this; }
friend modular pow(modular a, ll p) {
modular ans = 1; for (; p; p /= 2, a *= a) if (p&1) ans *= a;
return ans;
}
friend modular inv(const modular& a) {
auto i = invGeneral(a.val,MOD); assert(i != -1);
return i;
} // equivalent to return exp(b,MOD-2) if MOD is prime
modular& operator/=(const modular& m) { return (*this) *= inv(m); }
friend modular operator+(modular a, const modular& b) { return a += b; }
friend modular operator-(modular a, const modular& b) { return a -= b; }
friend modular operator*(modular a, const modular& b) { return a *= b; }
friend modular operator/(modular a, const modular& b) { return a /= b; }
};
typedef modular<int> mi;
typedef pair<mi,mi> pmi;
typedef vector<mi> vmi;
typedef vector<pmi> vpmi;
mt19937 rng(chrono::steady_clock::now().time_since_epoch().count());
typedef ll T;
namespace Point {
typedef pair<T,T> P;
typedef vector<P> vP;
P dir(T ang) {
auto c = exp(ang*complex<T>(0,1));
return P(c.real(),c.imag());
}
T norm(P x) { return x.f*x.f+x.s*x.s; }
T abs(P x) { return sqrt(norm(x)); }
T angle(P x) { return atan2(x.s,x.f); }
P conj(P x) { return P(x.f,-x.s); }
P operator+(const P& l, const P& r) { return P(l.f+r.f,l.s+r.s); }
P operator-(const P& l, const P& r) { return P(l.f-r.f,l.s-r.s); }
P operator*(const P& l, const T& r) { return P(l.f*r,l.s*r); }
P operator*(const T& l, const P& r) { return r*l; }
P operator/(const P& l, const T& r) { return P(l.f/r,l.s/r); }
P operator*(const P& l, const P& r) { return P(l.f*r.f-l.s*r.s,l.s*r.f+l.f*r.s); }
P operator/(const P& l, const P& r) { return l*conj(r)/norm(r); }
P& operator+=(P& l, const P& r) { return l = l+r; }
P& operator-=(P& l, const P& r) { return l = l-r; }
P& operator*=(P& l, const T& r) { return l = l*r; }
P& operator/=(P& l, const T& r) { return l = l/r; }
P& operator*=(P& l, const P& r) { return l = l*r; }
P& operator/=(P& l, const P& r) { return l = l/r; }
P unit(P x) { return x/abs(x); }
T dot(P a, P b) { return (conj(a)*b).f; }
T cross(P a, P b) { return (conj(a)*b).s; }
T cross(P p, P a, P b) { return cross(a-p,b-p); }
T dist(P p, P a, P b) { return std::abs(cross(p,a,b))/abs(a-b); }
P rotate(P a, T b) { return a*P(cos(b),sin(b)); }
P reflect(P p, P a, P b) { return a+conj((p-a)/(b-a))*(b-a); }
P foot(P p, P a, P b) { return (p+reflect(p,a,b))/(T)2; }
P extension(P a, P b, P c, P d) {
T x = cross(a,b,c), y = cross(a,b,d);
return (d*x-c*y)/(x-y);
}
// computes the intersection of line segments AB, CD
// verification: https://open.kattis.com/problems/segmentintersection
vP segIntersect(P a, P b, P c, P d) {
if (a > b) swap(a,b);
if (c > d) swap(c,d);
auto a1 = cross(a,b,c), a2 = cross(a,b,d);
if (a1 > a2) swap(a1,a2);
if (!(a1 <= 0 && a2 >= 0)) return {};
if (a1 == 0 && a2 == 0) {
if (cross(a,c,d) != 0) return {};
auto x1 = max(a,c), x2 = min(b,d);
if (x1 > x2) return {};
if (x1 == x2) return {x1};
return {x1,x2};
}
auto z = extension(a,b,c,d);
if (a <= z && z <= b) return {z};
return {};
}
// sorts points according to atan2
// verification: ?
template<class T> int half(pair<T,T> x) { return mp(x.s,x.f) > mp((T)0,(T)0); }
bool cmp(P a, P b) {
int A = half(a), B = half(b);
if (A != B) return A < B;
return cross(a,b) > 0;
}
// computes the center of mass of a polygon with constant mass per unit area
// verification: kattis polygonarea, VT HSPC 2018 Holiday Stars
T area(const vP& v) {
T area = 0;
F0R(i,sz(v)) {
int j = (i+1)%sz(v); T a = cross(v[i],v[j]);
area += a;
}
return std::abs(area)/2;
}
P centroid(const vP& v) {
P cen(0,0); T area = 0; // 2*signed area
F0R(i,sz(v)) {
int j = (i+1)%sz(v); T a = cross(v[i],v[j]);
cen += a*(v[i]+v[j]); area += a;
}
return cen/area/(T)3;
}
// tests whether a point is inside, on, or outside the perimeter of any polygon
// verification: https://open.kattis.com/problems/pointinpolygon
string inPoly(const vP& p, P z) {
int n = sz(p), ans = 0;
F0R(i,n) {
P x = p[i], y = p[(i+1)%n];
if (cross(x,y,z) == 0 && min(x,y) <= z && z <= max(x,y)) return "on";
if (x.s > y.s) swap(x,y);
if (x.s <= z.s && y.s > z.s && cross(z,x,y) > 0) ans ^= 1;
}
return ans ? "in" : "out";
}
pair<P,double> ccCenter(P a, P b, P c) { // circumcenter
b -= a; c -= a;
P res = b*c*(conj(c)-conj(b))/(b*conj(c)-conj(b)*c);
return {a+res,abs(res)};
}
pair<P, double> mec(vP ps) { // minimum enclosing circle, ex. USACO Camp 2019 Contest 2 #4
shuffle(all(ps), mt19937(time(0)));
P o = ps[0]; double r = 0, EPS = 1 + 1e-8;
F0R(i,sz(ps)) if (abs(o-ps[i]) > r*EPS) {
o = ps[i], r = 0;
F0R(j,i) if (abs(o-ps[j]) > r*EPS) {
o = (ps[i]+ps[j])/2, r = abs(o-ps[i]);
F0R(k,j) if (abs(o-ps[k]) > r*EPS)
tie(o,r) = ccCenter(ps[i],ps[j],ps[k]);
}
}
return {o,r};
}
};
using namespace Point;
vP convex_hull(vP P) {
sort(all(P)); P.erase(unique(all(P)),P.end());
int n = sz(P); if (n <= 1) return P;
vP up = {P[n-1]};
F0Rd(i,n-1) {
while (sz(up) > 1 && cross(up[sz(up)-2], up.back(), P[i]) <= 0) up.pop_back();
up.pb(P[i]);
}
while (sz(up) > 1 && up[sz(up)-1].s <= up[sz(up)-2].s) up.pop_back();
reverse(all(up));
F0R(i,sz(up)-1) assert(up[i].f < up[i+1].f && up[i].s > up[i+1].s);
return up;
}
map<P,int> mx, my;
int indx[300005], indy[300005], rx[300005], ry[300005];
vP x, y;
vP xx[2], yy[2];
void Init(std::vector<int> A, std::vector<int> D, std::vector<int> p){
int N = A.size();
F0R(i,N) {
x.pb(P(A[i],p[i])); mx[P(A[i],p[i])] = i;
y.pb(P(D[i],p[i])); my[P(D[i],p[i])] = i;
indx[i] = indy[i] = -1;
}
x = convex_hull(x), y = convex_hull(y);
F0R(i,sz(x)) {
int t = mx[x[i]];
indx[t] = i&1; rx[i] = t;
}
F0R(i,sz(y)) {
int t = my[y[i]];
indy[t] = i&1; ry[i] = t;
}
F0R(i,N) {
P z(A[i],p[i]);
if (indx[i] != 1) xx[0].pb(z);
if (indx[i] != 0) xx[1].pb(z);
z = P(D[i],p[i]);
if (indy[i] != 1) yy[0].pb(z);
if (indy[i] != 0) yy[1].pb(z);
}
F0R(i,2) {
xx[i] = convex_hull(xx[i]);
yy[i] = convex_hull(yy[i]);
}
ps("HUH",y,yy[0],yy[1]);
}
ll eval(P a, int X, int Y) {
return a.f*X+a.s*Y;
}
int bes(vP& a, int X, int Y) {
int lo = 0, hi = sz(a)-1;
while (lo < hi) {
int mid = (lo+hi)/2;
if (eval(a[mid],X,Y) < eval(a[mid+1],X,Y)) lo = mid+1;
else hi = mid;
}
return lo;
}
long long BestSquad(int X, int Y){
int a = bes(x,X,Y), b = bes(y,X,Y);
ll ea = eval(x[a],X,Y), eb = eval(y[b],X,Y);
if (rx[a] != ry[b]) return ea+eb;
ll ans1 = ea;
vP& v = yy[(b&1)^1];
ans1 += eval(v[bes(v,X,Y)],X,Y);
ll ans2 = eb;
v = xx[(a&1)^1];
ans2 += eval(v[bes(v,X,Y)],X,Y);
return max(ans1,ans2);
}
