#line 1 "library/my_template.hpp"
#if defined(LOCAL)
#include <my_template_compiled.hpp>
#else
#pragma GCC optimize("Ofast")
#pragma GCC optimize("unroll-loops")

#include <bits/stdc++.h>

using namespace std;

using ll = long long;
using u32 = unsigned int;
using u64 = unsigned long long;
using i128 = __int128;

template <class T>
constexpr T infty = 0;
template <>
constexpr int infty<int> = 1'000'000'000;
template <>
constexpr ll infty<ll> = ll(infty<int>) * infty<int> * 2;
template <>
constexpr u32 infty<u32> = infty<int>;
template <>
constexpr u64 infty<u64> = infty<ll>;
template <>
constexpr i128 infty<i128> = i128(infty<ll>) * infty<ll>;
template <>
constexpr double infty<double> = infty<ll>;
template <>
constexpr long double infty<long double> = infty<ll>;

using pi = pair<ll, ll>;
using vi = vector<ll>;
template <class T>
using vc = vector<T>;
template <class T>
using vvc = vector<vc<T>>;
template <class T>
using vvvc = vector<vvc<T>>;
template <class T>
using vvvvc = vector<vvvc<T>>;
template <class T>
using vvvvvc = vector<vvvvc<T>>;
template <class T>
using pq = priority_queue<T>;
template <class T>
using pqg = priority_queue<T, vector<T>, greater<T>>;

#define vv(type, name, h, ...) \
  vector<vector<type>> name(h, vector<type>(__VA_ARGS__))
#define vvv(type, name, h, w, ...)   \
  vector<vector<vector<type>>> name( \
      h, vector<vector<type>>(w, vector<type>(__VA_ARGS__)))
#define vvvv(type, name, a, b, c, ...)       \
  vector<vector<vector<vector<type>>>> name( \
      a, vector<vector<vector<type>>>(       \
             b, vector<vector<type>>(c, vector<type>(__VA_ARGS__))))

// https://trap.jp/post/1224/
#define FOR1(a) for (ll _ = 0; _ < ll(a); ++_)
#define FOR2(i, a) for (ll i = 0; i < ll(a); ++i)
#define FOR3(i, a, b) for (ll i = a; i < ll(b); ++i)
#define FOR4(i, a, b, c) for (ll i = a; i < ll(b); i += (c))
#define FOR1_R(a) for (ll i = (a)-1; i >= ll(0); --i)
#define FOR2_R(i, a) for (ll i = (a)-1; i >= ll(0); --i)
#define FOR3_R(i, a, b) for (ll i = (b)-1; i >= ll(a); --i)
#define overload4(a, b, c, d, e, ...) e
#define overload3(a, b, c, d, ...) d
#define FOR(...) overload4(__VA_ARGS__, FOR4, FOR3, FOR2, FOR1)(__VA_ARGS__)
#define FOR_R(...) overload3(__VA_ARGS__, FOR3_R, FOR2_R, FOR1_R)(__VA_ARGS__)

#define FOR_subset(t, s) \
  for (ll t = (s); t >= 0; t = (t == 0 ? -1 : (t - 1) & (s)))
#define all(x) x.begin(), x.end()
#define len(x) ll(x.size())
#define elif else if

#define eb emplace_back
#define mp make_pair
#define mt make_tuple
#define fi first
#define se second

#define stoi stoll

int popcnt(int x) { return __builtin_popcount(x); }
int popcnt(u32 x) { return __builtin_popcount(x); }
int popcnt(ll x) { return __builtin_popcountll(x); }
int popcnt(u64 x) { return __builtin_popcountll(x); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 1, 2)
int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(u32 x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
int topbit(u64 x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 0, 2)
int lowbit(int x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(u32 x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }
int lowbit(u64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }

template <typename T, typename U>
T ceil(T x, U y) {
  return (x > 0 ? (x + y - 1) / y : x / y);
}
template <typename T, typename U>
T floor(T x, U y) {
  return (x > 0 ? x / y : (x - y + 1) / y);
}
template <typename T, typename U>
pair<T, T> divmod(T x, U y) {
  T q = floor(x, y);
  return {q, x - q * y};
}

template <typename T, typename U>
T SUM(const vector<U> &A) {
  T sum = 0;
  for (auto &&a: A) sum += a;
  return sum;
}

#define MIN(v) *min_element(all(v))
#define MAX(v) *max_element(all(v))
#define LB(c, x) distance((c).begin(), lower_bound(all(c), (x)))
#define UB(c, x) distance((c).begin(), upper_bound(all(c), (x)))
#define UNIQUE(x) \
  sort(all(x)), x.erase(unique(all(x)), x.end()), x.shrink_to_fit()

template <typename T>
T POP(deque<T> &que) {
  T a = que.front();
  que.pop_front();
  return a;
}
template <typename T>
T POP(pq<T> &que) {
  T a = que.top();
  que.pop();
  return a;
}
template <typename T>
T POP(pqg<T> &que) {
  assert(!que.empty());
  T a = que.top();
  que.pop();
  return a;
}
template <typename T>
T POP(vc<T> &que) {
  assert(!que.empty());
  T a = que.back();
  que.pop_back();
  return a;
}

template <typename F>
ll binary_search(F check, ll ok, ll ng, bool check_ok = true) {
  if (check_ok) assert(check(ok));
  while (abs(ok - ng) > 1) {
    auto x = (ng + ok) / 2;
    tie(ok, ng) = (check(x) ? mp(x, ng) : mp(ok, x));
  }
  return ok;
}
template <typename F>
double binary_search_real(F check, double ok, double ng, int iter = 100) {
  FOR(iter) {
    double x = (ok + ng) / 2;
    tie(ok, ng) = (check(x) ? mp(x, ng) : mp(ok, x));
  }
  return (ok + ng) / 2;
}

template <class T, class S>
inline bool chmax(T &a, const S &b) {
  return (a < b ? a = b, 1 : 0);
}
template <class T, class S>
inline bool chmin(T &a, const S &b) {
  return (a > b ? a = b, 1 : 0);
}

// ? は -1
vc<int> s_to_vi(const string &S, char first_char) {
  vc<int> A(S.size());
  FOR(i, S.size()) { A[i] = (S[i] != '?' ? S[i] - first_char : -1); }
  return A;
}

template <typename T, typename U>
vector<T> cumsum(vector<U> &A, int off = 1) {
  int N = A.size();
  vector<T> B(N + 1);
  FOR(i, N) { B[i + 1] = B[i] + A[i]; }
  if (off == 0) B.erase(B.begin());
  return B;
}

// stable sort
template <typename T>
vector<int> argsort(const vector<T> &A) {
  vector<int> ids(len(A));
  iota(all(ids), 0);
  sort(all(ids),
       [&](int i, int j) { return (A[i] == A[j] ? i < j : A[i] < A[j]); });
  return ids;
}

// A[I[0]], A[I[1]], ...
template <typename T>
vc<T> rearrange(const vc<T> &A, const vc<int> &I) {
  vc<T> B(len(I));
  FOR(i, len(I)) B[i] = A[I[i]];
  return B;
}
#endif
#line 1 "library/other/io.hpp"
// based on yosupo's fastio
#include <unistd.h>

namespace fastio {
#define FASTIO
// クラスが read(), print() を持っているかを判定するメタ関数
struct has_write_impl {
  template <class T>
  static auto check(T &&x) -> decltype(x.write(), std::true_type{});

  template <class T>
  static auto check(...) -> std::false_type;
};

template <class T>
class has_write : public decltype(has_write_impl::check<T>(std::declval<T>())) {
};

struct has_read_impl {
  template <class T>
  static auto check(T &&x) -> decltype(x.read(), std::true_type{});

  template <class T>
  static auto check(...) -> std::false_type;
};

template <class T>
class has_read : public decltype(has_read_impl::check<T>(std::declval<T>())) {};

struct Scanner {
  FILE *fp;
  char line[(1 << 15) + 1];
  size_t st = 0, ed = 0;
  void reread() {
    memmove(line, line + st, ed - st);
    ed -= st;
    st = 0;
    ed += fread(line + ed, 1, (1 << 15) - ed, fp);
    line[ed] = '\0';
  }
  bool succ() {
    while (true) {
      if (st == ed) {
        reread();
        if (st == ed) return false;
      }
      while (st != ed && isspace(line[st])) st++;
      if (st != ed) break;
    }
    if (ed - st <= 50) {
      bool sep = false;
      for (size_t i = st; i < ed; i++) {
        if (isspace(line[i])) {
          sep = true;
          break;
        }
      }
      if (!sep) reread();
    }
    return true;
  }
  template <class T, enable_if_t<is_same<T, string>::value, int> = 0>
  bool read_single(T &ref) {
    if (!succ()) return false;
    while (true) {
      size_t sz = 0;
      while (st + sz < ed && !isspace(line[st + sz])) sz++;
      ref.append(line + st, sz);
      st += sz;
      if (!sz || st != ed) break;
      reread();
    }
    return true;
  }
  template <class T, enable_if_t<is_integral<T>::value, int> = 0>
  bool read_single(T &ref) {
    if (!succ()) return false;
    bool neg = false;
    if (line[st] == '-') {
      neg = true;
      st++;
    }
    ref = T(0);
    while (isdigit(line[st])) { ref = 10 * ref + (line[st++] & 0xf); }
    if (neg) ref = -ref;
    return true;
  }
  template <typename T,
            typename enable_if<has_read<T>::value>::type * = nullptr>
  inline bool read_single(T &x) {
    x.read();
    return true;
  }
  bool read_single(double &ref) {
    string s;
    if (!read_single(s)) return false;
    ref = std::stod(s);
    return true;
  }
  bool read_single(char &ref) {
    string s;
    if (!read_single(s) || s.size() != 1) return false;
    ref = s[0];
    return true;
  }
  template <class T>
  bool read_single(vector<T> &ref) {
    for (auto &d: ref) {
      if (!read_single(d)) return false;
    }
    return true;
  }
  template <class T, class U>
  bool read_single(pair<T, U> &p) {
    return (read_single(p.first) && read_single(p.second));
  }
  template <size_t N = 0, typename T>
  void read_single_tuple(T &t) {
    if constexpr (N < std::tuple_size<T>::value) {
      auto &x = std::get<N>(t);
      read_single(x);
      read_single_tuple<N + 1>(t);
    }
  }
  template <class... T>
  bool read_single(tuple<T...> &tpl) {
    read_single_tuple(tpl);
    return true;
  }
  void read() {}
  template <class H, class... T>
  void read(H &h, T &... t) {
    bool f = read_single(h);
    assert(f);
    read(t...);
  }
  Scanner(FILE *fp) : fp(fp) {}
};

struct Printer {
  Printer(FILE *_fp) : fp(_fp) {}
  ~Printer() { flush(); }

  static constexpr size_t SIZE = 1 << 15;
  FILE *fp;
  char line[SIZE], small[50];
  size_t pos = 0;
  void flush() {
    fwrite(line, 1, pos, fp);
    pos = 0;
  }
  void write(const char val) {
    if (pos == SIZE) flush();
    line[pos++] = val;
  }
  template <class T, enable_if_t<is_integral<T>::value, int> = 0>
  void write(T val) {
    if (pos > (1 << 15) - 50) flush();
    if (val == 0) {
      write('0');
      return;
    }
    if (val < 0) {
      write('-');
      val = -val; // todo min
    }
    size_t len = 0;
    while (val) {
      small[len++] = char(0x30 | (val % 10));
      val /= 10;
    }
    for (size_t i = 0; i < len; i++) { line[pos + i] = small[len - 1 - i]; }
    pos += len;
  }
  void write(const string s) {
    for (char c: s) write(c);
  }
  void write(const char *s) {
    size_t len = strlen(s);
    for (size_t i = 0; i < len; i++) write(s[i]);
  }
  void write(const double x) {
    ostringstream oss;
    oss << fixed << setprecision(15) << x;
    string s = oss.str();
    write(s);
  }
  void write(const long double x) {
    ostringstream oss;
    oss << fixed << setprecision(15) << x;
    string s = oss.str();
    write(s);
  }
  template <typename T,
            typename enable_if<has_write<T>::value>::type * = nullptr>
  inline void write(T x) {
    x.write();
  }
  template <class T>
  void write(const vector<T> val) {
    auto n = val.size();
    for (size_t i = 0; i < n; i++) {
      if (i) write(' ');
      write(val[i]);
    }
  }
  template <class T, class U>
  void write(const pair<T, U> val) {
    write(val.first);
    write(' ');
    write(val.second);
  }
  template <size_t N = 0, typename T>
  void write_tuple(const T t) {
    if constexpr (N < std::tuple_size<T>::value) {
      if constexpr (N > 0) { write(' '); }
      const auto x = std::get<N>(t);
      write(x);
      write_tuple<N + 1>(t);
    }
  }
  template <class... T>
  bool write(tuple<T...> tpl) {
    write_tuple(tpl);
    return true;
  }
  template <class T, size_t S>
  void write(const array<T, S> val) {
    auto n = val.size();
    for (size_t i = 0; i < n; i++) {
      if (i) write(' ');
      write(val[i]);
    }
  }
  void write(i128 val) {
    string s;
    bool negative = 0;
    if (val < 0) {
      negative = 1;
      val = -val;
    }
    while (val) {
      s += '0' + int(val % 10);
      val /= 10;
    }
    if (negative) s += "-";
    reverse(all(s));
    if (len(s) == 0) s = "0";
    write(s);
  }
};
Scanner scanner = Scanner(stdin);
Printer printer = Printer(stdout);
void flush() { printer.flush(); }
void print() { printer.write('\n'); }
template <class Head, class... Tail>
void print(Head &&head, Tail &&... tail) {
  printer.write(head);
  if (sizeof...(Tail)) printer.write(' ');
  print(forward<Tail>(tail)...);
}

void read() {}
template <class Head, class... Tail>
void read(Head &head, Tail &... tail) {
  scanner.read(head);
  read(tail...);
}
} // namespace fastio
using fastio::print;
using fastio::flush;
using fastio::read;

#define INT(...)   \
  int __VA_ARGS__; \
  read(__VA_ARGS__)
#define LL(...)   \
  ll __VA_ARGS__; \
  read(__VA_ARGS__)
#define STR(...)      \
  string __VA_ARGS__; \
  read(__VA_ARGS__)
#define CHAR(...)   \
  char __VA_ARGS__; \
  read(__VA_ARGS__)
#define DBL(...)      \
  double __VA_ARGS__; \
  read(__VA_ARGS__)

#define VEC(type, name, size) \
  vector<type> name(size);    \
  read(name)
#define VV(type, name, h, w)                     \
  vector<vector<type>> name(h, vector<type>(w)); \
  read(name)

void YES(bool t = 1) { print(t ? "YES" : "NO"); }
void NO(bool t = 1) { YES(!t); }
void Yes(bool t = 1) { print(t ? "Yes" : "No"); }
void No(bool t = 1) { Yes(!t); }
void yes(bool t = 1) { print(t ? "yes" : "no"); }
void no(bool t = 1) { yes(!t); }
#line 3 "main.cpp"

#line 1 "library/nt/rational.hpp"
template <typename T = long long, bool REDUCE = true>
struct Rational {
  T num, den;

  Rational() : num(0), den(1) {}
  Rational(T x) : num(x), den(1) {}
  Rational(T a, T b, bool coprime = false) : num(a), den(b) {
    if (!coprime && REDUCE) reduce();
  }

  static T gcd(T a, T b) {
    a = max(a, -a), b = max(b, -b);
    while (b) {
      a %= b;
      swap(a, b);
    }
    return a;
  }

  void reduce() {
    if (!REDUCE) return;
    T g = gcd(num, den);
    num /= g, den /= g;
  }

  Rational &operator+=(const Rational &p) {
    T g = (REDUCE ? gcd(den, p.den) : 1);
    num = num * (p.den / g) + p.num * (den / g);
    den *= p.den / g;
    reduce();
    return *this;
  }
  Rational &operator-=(const Rational &p) {
    T g = (REDUCE ? gcd(den, p.den) : 1);
    num = num * (p.den / g) - p.num * (den / g);
    den *= p.den / g;
    reduce();
    return *this;
  }
  Rational &operator*=(const Rational &p) {
    T g1 = (REDUCE ? gcd(num, p.den) : 1);
    T g2 = (REDUCE ? gcd(den, p.num) : 1);
    num = (num / g1) * (p.num / g2);
    den = (den / g2) * (p.den / g1);
    return *this;
  }
  Rational &operator/=(const Rational &p) {
    T g1 = (REDUCE ? gcd(num, p.num) : 1);
    T g2 = (REDUCE ? gcd(den, p.den) : 1);
    num = (num / g1) * (p.den / g2);
    den = (den / g2) * (p.num / g1);
    if (den < 0) num = -num, den = -den;
    return *this;
  }

  Rational operator-() const { return Rational(-num, den); }
  Rational operator+(const Rational &p) const { return Rational(*this) += p; }
  Rational operator-(const Rational &p) const { return Rational(*this) -= p; }
  Rational operator*(const Rational &p) const { return Rational(*this) *= p; }
  Rational operator/(const Rational &p) const { return Rational(*this) /= p; }
  bool operator==(const Rational &p) const {
    return num * p.den == p.num * den;
  }
  bool operator!=(const Rational &p) const {
    return num * p.den != p.num * den;
  }
  bool operator<(const Rational &p) const { return num * p.den < p.num * den; }
  bool operator>(const Rational &p) const { return num * p.den > p.num * den; }
  bool operator<=(const Rational &p) const {
    return num * p.den <= p.num * den;
  }
  bool operator>=(const Rational &p) const {
    return num * p.den >= p.num * den;
  }

  string to_string() { return std::to_string(num) + "/" + std::to_string(den); }
  double to_double() { return double(num) / double(den); }
};
#line 2 "library/geo/base.hpp"
template <typename T>
struct Point {
  T x, y;

  Point() = default;

  template <typename A, typename B>
  Point(A x, B y) : x(x), y(y) {}

  template <typename A, typename B>
  Point(pair<A, B> p) : x(p.fi), y(p.se) {}

  Point operator+(Point p) const { return {x + p.x, y + p.y}; }
  Point operator-(Point p) const { return {x - p.x, y - p.y}; }
  bool operator==(Point p) const { return x == p.x && y == p.y; }
  Point operator-() const { return {-x, -y}; }

  bool operator<(Point p) const {
    if (x != p.x) return x < p.x;
    return y < p.y;
  }
  T dot(Point other) { return x * other.x + y * other.y; }
  T det(Point other) { return x * other.y - y * other.x; }

  void read() { fastio::read(x), fastio::read(y); }
  void write() { fastio::printer.write(pair<T, T>({x, y})); }
};

// A -> B -> C と進むときに、左に曲がるならば +1、右に曲がるならば -1
template <typename T>
int ccw(Point<T> A, Point<T> B, Point<T> C) {
  T x = (B - A).det(C - A);
  if (x > 0) return 1;
  if (x < 0) return -1;
  return 0;
}

template <typename REAL, typename T>
REAL dist(Point<T> A, Point<T> B) {
  A = A - B;
  T p = A.dot(A);
  return sqrt(REAL(p));
}

template <typename T>
struct Line {
  T a, b, c;

  Line(T a, T b, T c) : a(a), b(b), c(c) {}
  Line(Point<T> A, Point<T> B) {
    a = A.y - B.y, b = B.x - A.x, c = A.x * B.y - A.y * B.x;
  }
  Line(T x1, T y1, T x2, T y2) : Line(Point<T>(x1, y1), Point<T>(x2, y2)) {}

  template <typename U>
  U eval(Point<U> P) {
    return a * P.x + b * P.y + c;
  }

  template <typename U>
  T eval(U x, U y) {
    return a * x + b * y + c;
  }

  bool is_parallel(Line other) { return a * other.b - b * other.a == 0; }

  bool is_orthogonal(Line other) { return a * other.a + b * other.b == 0; }
};

template <typename T>
struct Segment {
  Point<T> A, B;

  Segment(Point<T> A, Point<T> B) : A(A), B(B) {}
  Segment(T x1, T y1, T x2, T y2)
      : Segment(Point<T>(x1, y1), Point<T>(x2, y2)) {}

  template <enable_if_t<is_integral<T>::value, int> = 0>
  bool contain(Point<T> C) {
    T det = (C - A).det(B - A);
    if (det != 0) return 0;
    return (C - A).dot(B - A) >= 0 && (C - B).dot(A - B) >= 0;
  }

  Line<T> to_Line() { return Line(A, B); }
};

template <typename REAL>
struct Circle {
  Point<REAL> O;
  REAL r;
  Circle(Point<REAL> O, REAL r) : O(O), r(r) {}
  Circle(REAL x, REAL y, REAL r) : O(x, y), r(r) {}
  template <typename T>
  bool contain(Point<T> p) {
    REAL dx = p.x - O.x, dy = p.y - O.y;
    return dx * dx + dy * dy <= r * r;
  }
};

template <typename T>
struct Polygon {
  vc<Point<T>> points;
  T a;

  template <typename A, typename B>
  Polygon(vc<pair<A, B>> pairs) {
    for (auto&& [a, b]: pairs) points.eb(Point<T>(a, b));
    build();
  }
  Polygon(vc<Point<T>> points) : points(points) { build(); }

  int size() { return len(points); }

  template <typename REAL>
  REAL area() {
    return a * 0.5;
  }

  template <enable_if_t<is_integral<T>::value, int> = 0>
  T area_2() {
    return a;
  }

  bool is_convex() {
    FOR(j, len(points)) {
      int i = (j == 0 ? len(points) - 1 : j - 1);
      int k = (j == len(points) - 1 ? 0 : j + 1);
      if ((points[j] - points[i]).det(points[k] - points[j]) < 0) return false;
    }
    return true;
  }

private:
  void build() {
    a = 0;
    FOR(i, len(points)) {
      int j = (i + 1 == len(points) ? 0 : i + 1);
      a += points[i].det(points[j]);
    }
    if (a < 0) {
      a = -a;
      reverse(all(points));
    }
  }
};
#line 2 "library/geo/convex_hull.hpp"

template <typename T>
vector<int> ConvexHull(vector<pair<T, T>>& XY, string mode = "full",
                       bool inclusive = false, bool sorted = false) {
  assert(mode == "full" || mode == "lower" || mode == "upper");
  ll N = XY.size();
  if (N == 1) return {0};
  if (N == 2) return {0, 1};
  vc<int> I = argsort(XY);

  auto check = [&](ll i, ll j, ll k) -> bool {
    auto xi = XY[i].fi, yi = XY[i].se;
    auto xj = XY[j].fi, yj = XY[j].se;
    auto xk = XY[k].fi, yk = XY[k].se;
    auto dx1 = xj - xi, dy1 = yj - yi;
    auto dx2 = xk - xj, dy2 = yk - yj;
    T det = dx1 * dy2 - dy1 * dx2;
    return (inclusive ? det >= 0 : det > 0);
  };

  auto calc = [&]() {
    vector<int> P;
    for (auto&& k: I) {
      while (P.size() > 1) {
        auto i = P[P.size() - 2];
        auto j = P[P.size() - 1];
        if (check(i, j, k)) break;
        P.pop_back();
      }
      P.eb(k);
    }
    return P;
  };

  vc<int> P;
  if (mode == "full" || mode == "lower") {
    vc<int> Q = calc();
    P.insert(P.end(), all(Q));
  }
  if (mode == "full" || mode == "upper") {
    if (!P.empty()) P.pop_back();
    reverse(all(I));
    vc<int> Q = calc();
    P.insert(P.end(), all(Q));
  }
  if (mode == "upper") reverse(all(P));
  if (len(P) >= 2 && P[0] == P.back()) P.pop_back();
  return P;
}

template <typename T>
vector<int> ConvexHull(vector<Point<T>>& XY, string mode = "full",
                       bool inclusive = false, bool sorted = false) {
  assert(mode == "full" || mode == "lower" || mode == "upper");
  ll N = XY.size();
  if (N == 1) return {0};
  if (N == 2) return {0, 1};
  vc<int> I = argsort(XY);

  auto check = [&](ll i, ll j, ll k) -> bool {
    auto xi = XY[i].x, yi = XY[i].y;
    auto xj = XY[j].x, yj = XY[j].y;
    auto xk = XY[k].x, yk = XY[k].y;
    auto dx1 = xj - xi, dy1 = yj - yi;
    auto dx2 = xk - xj, dy2 = yk - yj;
    T det = dx1 * dy2 - dy1 * dx2;
    return (inclusive ? det >= 0 : det > 0);
  };

  auto calc = [&]() {
    vector<int> P;
    for (auto&& k: I) {
      while (P.size() > 1) {
        auto i = P[P.size() - 2];
        auto j = P[P.size() - 1];
        if (check(i, j, k)) break;
        P.pop_back();
      }
      P.eb(k);
    }
    return P;
  };

  vc<int> P;
  if (mode == "full" || mode == "lower") {
    vc<int> Q = calc();
    P.insert(P.end(), all(Q));
  }
  if (mode == "full" || mode == "upper") {
    if (!P.empty()) P.pop_back();
    reverse(all(I));
    vc<int> Q = calc();
    P.insert(P.end(), all(Q));
  }
  if (mode == "upper") reverse(all(P));
  if (len(P) >= 2 && P[0] == P.back()) P.pop_back();
  return P;
}
#line 1 "library/geo/convex_polygon.hpp"

template <typename T>
struct ConvexPolygon {
  using P = Point<T>;
  int n;
  vc<P> point;

  ConvexPolygon(vc<P> point) : n(len(point)), point(point) { assert(n >= 1); }

  // 比較関数 comp(i,j)
  template <typename F>
  int periodic_min_comp(F comp) {
    int L = 0, M = n, R = n + n;
    while (1) {
      if (R - L == 2) break;
      int L1 = (L + M) / 2, R1 = (M + R + 1) / 2;
      if (comp(L1, M)) { R = M, M = L1; }
      elif (comp(R1, M)) { L = M, M = R1; }
      else {
        L = L1, R = R1;
      }
    }
    return M % n;
  }

  int nxt_idx(int i) { return (i + 1 == n ? 0 : i + 1); }
  int prev_idx(int i) { return (i == 0 ? n - 1 : i - 1); }

  // 中：1, 境界：0, 外：-1
  // int side(P p) {}

  pair<int, T> min_dot(P p) {
    int idx = periodic_min_comp([&](int i, int j) -> bool {
      return point[i % n].dot(p) < point[j % n].dot(p);
    });
    return {idx, point[idx].dot(p)};
  }
  pair<int, T> max_dot(P p) {
    int idx = periodic_min_comp([&](int i, int j) -> bool {
      return point[i % n].dot(p) > point[j % n].dot(p);
    });
    return {idx, point[idx].dot(p)};
  }
  // pair<int, int> visible_range(P p) {}
};
#line 7 "main.cpp"

using Re = double;

using QQ = Rational<i128, false>;
using Poly = ConvexPolygon<QQ>;

void solve() {
  LL(N);
  VEC(pi, AB, N);
  ll O = 0;
  {
    vc<pi> tmp;
    for (auto&& [a, b]: AB) {
      if (a == 0 && b == 0) {
        ++O;
      } else {
        tmp.eb(a, b);
      }
    }
    swap(AB, tmp);
    N = len(AB);
  }

  using P = Point<QQ>;
  vc<P> point(N);
  FOR(i, N) {
    auto [a, b] = AB[i];
    point[i].x = QQ{a, a * a + b * b};
    point[i].y = QQ{b, a * a + b * b};
  }

  using Poly = ConvexPolygon<Re>;
  vc<Poly> hull;
  FOR(5) {
    N = len(point);
    if (point.empty()) break;
    vc<int> I = ConvexHull<QQ>(point, "full");
    vc<bool> use(N);
    for (auto&& i: I) use[i] = 1;
    vc<Point<Re>> polygon;
    for (auto&& idx: I) {
      polygon.eb(point[idx].x.to_double(), point[idx].y.to_double());
    }
    vc<P> rest;
    FOR(i, N) {
      if (!use[i]) rest.eb(point[i]);
    }
    hull.eb(ConvexPolygon<Re>(polygon));
    swap(point, rest);
  }

  auto solve = [&](ll x, ll y, ll k) -> void {
    if (x == 0 && y == 0) return print(-1);
    if (k < O) { return print(0.0); }
    k -= O;
    vc<Re> ts;
    Point<Re> p = {x, y};
    for (auto&& poly: hull) {
      int idx = poly.max_dot(p).fi;
      vc<int> I;
      int i = idx;
      FOR(5) {
        I.eb(i);
        i = poly.prev_idx(i);
      }
      i = idx;
      FOR(5) {
        I.eb(i);
        i = poly.nxt_idx(i);
      }
      UNIQUE(I);
      for (auto&& idx: I) {
        Re dot = p.dot(poly.point[idx]);
        if (dot <= 0) continue;
        ts.eb(dot);
      }
    }
    sort(all(ts));
    reverse(all(ts));
    if (len(ts) <= k) { return print(-1); }
    Re ANS = 1.0 / ts[k];
    Re d = sqrtl(x * x + y * y);
    ANS *= d / 2;
    print(ANS);
  };

  LL(Q);
  FOR(Q) {
    LL(x, y, k);
    --k;
    solve(x, y, k);
  }
}

signed main() {
  solve();
  return 0;
}