// This is free and unencumbered software released into the public domain.

// Anyone is free to copy, modify, publish, use, compile, sell, or
// distribute this software, either in source code form or as a compiled
// binary, for any purpose, commercial or non-commercial, and by any
// means.

// In jurisdictions that recognize copyright laws, the author or authors
// of this software dedicate any and all copyright interest in the
// software to the public domain. We make this dedication for the benefit
// of the public at large and to the detriment of our heirs and
// successors. We intend this dedication to be an overt act of
// relinquishment in perpetuity of all present and future rights to this
// software under copyright law.

// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
// MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
// IN NO EVENT SHALL THE AUTHORS BE LIABLE FOR ANY CLAIM, DAMAGES OR
// OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
// ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
// OTHER DEALINGS IN THE SOFTWARE.

// For more information, please refer to <http://unlicense.org>

/****************/
/* template.hpp */
/****************/

#include <algorithm>
#include <cassert>
#include <functional>
#include <iomanip>
#include <iostream>
#include <limits>

using std::cerr;
using std::cout;
using std::endl;
using std::max;
using std::min;
using std::swap;

struct BoolName : std::numpunct<char> {
  std::string t, f;
  BoolName(std::string t, std::string f) : t(t), f(f) {}
  std::string do_truename() const { return t; }
  std::string do_falsename() const { return f; }
};

void setBoolName(std::string t, std::string f) {
  cout.imbue(std::locale(cout.getloc(), new BoolName(t, f)));
}

struct Initializer {
  Initializer() {
    cout << std::fixed << std::setprecision(15) << std::boolalpha;
    setBoolName("Yes", "No");
  }
} initializer;

struct Input {
  bool eof;

  Input() : eof(false) {}

  operator char() {
    char v;
    while (!(this->eof = (std::scanf("%c", &v) != 1)) && std::isspace(v)) {
    }
    return v;
  }

  operator int() {
    int v;
    this->eof = (std::scanf("%d", &v) != 1);
    return v;
  }

  operator long() {
    long v;
    this->eof = (std::scanf("%ld", &v) != 1);
    return v;
  }

  operator long long() {
    long long v;
    this->eof = (std::scanf("%lld", &v) != 1);
    return v;
  }

  operator unsigned int() {
    unsigned int v;
    this->eof = (std::scanf("%u", &v) != 1);
    return v;
  }

  operator unsigned long() {
    unsigned long v;
    this->eof = (std::scanf("%lu", &v) != 1);
    return v;
  }

  operator unsigned long long() {
    unsigned long long v;
    this->eof = (std::scanf("%llu", &v) != 1);
    return v;
  }

  operator double() {
    double v;
    this->eof = (std::scanf("%lf", &v) != 1);
    return v;
  }

  operator long double() {
    long double v;
    this->eof = (std::scanf("%Lf", &v) != 1);
    return v;
  }

  void ignore() const { getchar(); }
} in;

template <typename T> T abs(T a) { return a >= 0 ? a : -a; }

template <typename T, typename S> bool chmin(T &a, const S &b) {
  return a > b ? a = b, true : false;
}

template <typename T, typename S> bool chmax(T &a, const S &b) {
  return a < b ? a = b, true : false;
}

template <typename T, typename S> std::function<S(T)> cast() {
  return [](const T &t) { return static_cast<S>(t); };
}

template <typename T> T copy(const T &a) { return T(a); }

class ZeroPadding {
public:
  ZeroPadding(int n) : n(n) {}

  int n;
};

std::ostream &operator<<(std::ostream &os, const ZeroPadding &z) {
  os << std::setw(z.n) << std::setfill('0');
  return os;
}

template <typename T> constexpr T inf() {
  return std::numeric_limits<T>::max() / 2 - 1;
}

/******************/
/* arithmetic.hpp */
/******************/

template <typename T> class Addition {
public:
  template <typename V> T operator+(const V &v) const {
    return T(static_cast<const T &>(*this)) += v;
  }

  T operator++() { return static_cast<T &>(*this) += 1; }
};

template <typename T> class Subtraction {
public:
  template <typename V> T operator-(const V &v) const {
    return T(static_cast<const T &>(*this)) -= v;
  }
};

template <typename T> class Multiplication {
public:
  template <typename V> T operator*(const V &v) const {
    return T(static_cast<const T &>(*this)) *= v;
  }
};

template <typename T> class Division {
public:
  template <typename V> T operator/(const V &v) const {
    return T(static_cast<const T &>(*this)) /= v;
  }
};

template <typename T> class Modulus {
public:
  template <typename V> T operator%(const V &v) const {
    return T(static_cast<const T &>(*this)) %= v;
  }
};

template <typename T>
class IndivisibleArithmetic : public Addition<T>,
                              public Subtraction<T>,
                              public Multiplication<T> {};

template <typename T>
class Arithmetic : public IndivisibleArithmetic<T>, public Division<T> {};
/*********************/
/* bit_operation.hpp */
/*********************/

template <typename T> int least_bit(T n) {
  static_assert(sizeof(T) == 4 || sizeof(T) == 8, "unsupported size");
  if (sizeof(T) == 4) {
    return __builtin_ffs(n) - 1;
  }
  if (sizeof(T) == 8) {
    return __builtin_ffsll(n) - 1;
  }
}

// n must be greater than 0.
template <typename T> int least_bit_fast(T n) {
  static_assert(sizeof(T) == 4 || sizeof(T) == 8, "unsupported size");
  if (sizeof(T) == 4) {
    return __builtin_ctz(n);
  }
  if (sizeof(T) == 8) {
    return __builtin_ctzll(n);
  }
}

template <typename T> int most_bit(T n) {
  static_assert(sizeof(T) == 4 || sizeof(T) == 8, "unsupported size");
  if (sizeof(T) == 4) {
    return n ? 31 - __builtin_clz(n) : -1;
  }
  if (sizeof(T) == 8) {
    return n ? 63 - __builtin_clzll(n) : -1;
  }
}

template <typename T> int count_bit(T n) {
  static_assert(sizeof(T) == 4 || sizeof(T) == 8, "unsupported size");
  if (sizeof(T) == 4) {
    return __builtin_popcount(n);
  }
  if (sizeof(T) == 8) {
    return __builtin_popcountll(n);
  }
}

template <typename T> int bit_parity(T n) {
  static_assert(sizeof(T) == 4 || sizeof(T) == 8, "unsupported size");
  if (sizeof(T) == 4) {
    return __builtin_parity(n);
  }
  if (sizeof(T) == 8) {
    return __builtin_parityll(n);
  }
}
/*************/
/* tuple.hpp */
/*************/

#include <tuple>

template <typename... T> class Tuple : public std::tuple<T...> {
public:
  Tuple(Input &in) : std::tuple<T...>() { (void)in; }
};

template <typename T, typename... S>
class Tuple<T, S...> : public std::tuple<T, S...> {
public:
  Tuple() : std::tuple<T, S...>() {}

  Tuple(T t, S... s) : std::tuple<T, S...>(t, s...) {}

  Tuple(const std::tuple<T, S...> &t) : std::tuple<T, S...>(t) {}

  Tuple(Input &in) {
    auto a = std::tuple<T>(in);
    std::tuple<S...> b = Tuple<S...>(in);
    std::tuple<T, S...> c = std::tuple_cat(a, b);
    *this = c;
  }

  template <int n> auto &get() { return std::get<n>(*this); }

  template <int n> const auto &get() const { return std::get<n>(*this); }
};

template <typename... T> Tuple<T...> makeTuple(const T &... args) {
  return Tuple<T...>(args...);
}

namespace std {
template <typename... T>
class tuple_size<Tuple<T...>>
    : public std::integral_constant<size_t, sizeof...(T)> {};
template <std::size_t I, typename... T> class tuple_element<I, Tuple<T...>> {
public:
  using type = tuple_element_t<I, std::tuple<T...>>;
};
} // namespace std

/*****************/
/* container.hpp */
/*****************/

#include <vector>

template <typename T> class Container : public T {
private:
  using S = typename T::value_type;
  using Itr = typename T::iterator;

public:
  Container() : T() {}

  Container(int n) : T(n) {}

  Container(int n, S s) : T(n, s) {}

  template <typename Itr> Container(Itr first, Itr last) : T(first, last) {}

  Container(const std::initializer_list<S> &v) : T(v) {}

  Container(int n, Input &in) {
    std::vector<S> v(n);
    for (auto &i : v) {
      i = in;
    }
    *this = Container<T>(v.begin(), v.end());
  }

  S max() const { return *std::max_element(this->begin(), this->end()); }

  template <typename Function> auto max(Function func) const {
    std::vector<std::pair<decltype(func(S())), S>> res;
    for (const auto &i : *this) {
      res.emplace_back(func(i), i);
    }
    return std::max_element(res.begin(), res.end())->second;
  }

  S min() const { return *std::min_element(this->begin(), this->end()); }

  Tuple<S, S> minmax() const {
    auto itrs = std::minmax_element(this->begin(), this->end());
    return Tuple<S, S>(*itrs.first, *itrs.second);
  }

  template <typename Function> auto min(Function func) const {
    std::vector<std::pair<decltype(func(S())), S>> res;
    for (const auto &i : *this) {
      res.emplace_back(func(i), i);
    }
    return std::min_element(res.begin(), res.end())->second;
  }

  int argmax() const {
    return std::distance(this->begin(),
                         std::max_element(this->begin(), this->end()));
  }

  int argmin() const {
    return std::distance(this->begin(),
                         std::min_element(this->begin(), this->end()));
  }

  int find(const S &a) const {
    return std::distance(this->begin(),
                         std::find(this->begin(), this->end(), a));
  }

  bool contains(const S &a) const {
    return std::find(this->begin(), this->end(), a) != this->end();
  }

  int size() const { return T::size(); }

  std::pair<Itr, Itr> equal_range(const S &a) {
    return std::equal_range(this->begin(), this->end(), a);
  }

  template <typename Function> bool all_of(Function func) const {
    return std::all_of(this->begin(), this->end(), func);
  }

  template <typename Function> bool any_of(Function func) const {
    return std::any_of(this->begin(), this->end(), func);
  }

  template <typename Function> bool none_of(Function func) const {
    return std::none_of(this->begin(), this->end(), func);
  }

  int count(const S &s) const {
    return std::count(this->begin(), this->end(), s);
  }

  bool is_sorted() const { return std::is_sorted(this->begin(), this->end()); }

  void output(std::string sep = "\n", std::string end = "\n") const {
    bool first = true;
    for (const auto &i : *this) {
      if (!first) {
        cout << sep;
      }
      first = false;
      cout << i;
    }
    cout << end;
  }
};

/***********/
/* map.hpp */
/***********/

#include <map>

template <typename T, typename S> class Map : public Container<std::map<T, S>> {
public:
  Map() : Container<std::map<T, S>>() {}

  bool contains(const T &a) const { return this->count(a) != 0; }

  int count(const T &t) const { return std::map<T, S>::count(t); }
};
/***************/
/* ordered.hpp */
/***************/

template <typename T> class Ordered {
public:
  template <typename V> bool operator==(const V &v) const {
    return !(static_cast<T>(v) < static_cast<const T &>(*this) ||
             static_cast<const T &>(*this) < static_cast<T>(v));
  }

  template <typename V> bool operator!=(const V &v) const {
    return static_cast<T>(v) < static_cast<const T &>(*this) ||
           static_cast<const T &>(*this) < static_cast<T>(v);
  }

  template <typename V> bool operator>(const V &v) const {
    return static_cast<T>(v) < static_cast<const T &>(*this);
  }

  template <typename V> bool operator<=(const V &v) const {
    return !(static_cast<T>(v) < static_cast<const T &>(*this));
  }

  template <typename V> bool operator>=(const V &v) const {
    return !(static_cast<const T &>(*this) < static_cast<T>(v));
  }
};

/**************/
/* vector.hpp */
/**************/

#include <numeric>

template <typename T>
class Vector : public Container<std::vector<T>>, public Ordered<Vector<T>> {
public:
  Vector() = default;

  Vector(const Vector<T> &v) = default;

  Vector(int n) : Container<std::vector<T>>(n) {}

  Vector(int n, T t) : Container<std::vector<T>>(n, t) {}

  template <typename Itr>
  Vector(Itr first, Itr last) : Container<std::vector<T>>(first, last) {}

  Vector(const std::initializer_list<T> &v) : Container<std::vector<T>>(v) {}

  Vector(int n, Input &in) : Container<std::vector<T>>(n, in) {}

  Vector &operator+=(const Vector &v) {
    if (this->size() < v.size()) {
      this->resize(v.size());
    }
    for (int i = 0; i < v.size(); ++i) {
      (*this)[i] += v[i];
    }
    return *this;
  }

  Vector &operator+=(const T &v) {
    for (auto &i : *this) {
      i += v;
    }
    return *this;
  }

  Vector &operator-=(const Vector &v) {
    if (this->size() < v.size()) {
      this->resize(v.size());
    }
    for (int i = 0; i < v.size(); ++i) {
      (*this)[i] -= v[i];
    }
    return *this;
  }

  Vector &operator-=(const T &v) {
    for (auto &i : *this) {
      i -= v;
    }
    return *this;
  }

  Vector &operator*=(const Vector &v) {
    for (int i = 0; i < this->size(); ++i) {
      (*this)[i] *= v[i];
    }
    return *this;
  }

  Vector &operator*=(const T &v) {
    for (auto &i : *this) {
      i *= v;
    }
    return *this;
  }

  Vector &operator/=(const Vector &v) {
    for (int i = 0; i < this->size(); ++i) {
      (*this)[i] /= v[i];
    }
    return *this;
  }

  Vector &operator/=(const T &v) {
    for (auto &i : *this) {
      i /= v;
    }
    return *this;
  }

  Vector &operator%=(const Vector &v) {
    for (int i = 0; i < this->size(); ++i) {
      (*this)[i] %= v[i];
    }
    return *this;
  }

  Vector &operator%=(const T &v) {
    for (auto &i : *this) {
      i %= v;
    }
    return *this;
  }

  Vector operator+(const Vector &v) const { return Vector(*this) += v; }

  Vector operator+(const T &v) const { return Vector(*this) += v; }

  Vector operator-(const Vector &v) const { return Vector(*this) -= v; }

  Vector operator-(const T &v) const { return Vector(*this) -= v; }

  Vector operator*(const Vector &v) const { return Vector(*this) *= v; }

  Vector operator*(const T &v) const { return Vector(*this) *= v; }

  Vector operator/(const Vector &v) const { return Vector(*this) /= v; }

  Vector operator/(const T &v) const { return Vector(*this) /= v; }

  Vector operator%(const Vector &v) const { return Vector(*this) %= v; }

  Vector operator%(const T &v) const { return Vector(*this) %= v; }

  bool operator<(const Vector &v) const {
    if (this->size() != v.size()) {
      return this->size() < v.size();
    }
    for (int i = 0; i < this->size(); ++i) {
      if ((*this)[i] != v[i]) {
        return (*this)[i] < v[i];
      }
    }
    return false;
  }

  Vector operator-() const { return *this * -1; }

  T inner_product(const Vector<T> &v) const {
    return std::inner_product(this->begin(), this->end(), v.begin(), T(0));
  }

  Vector<T> &partial_sort(int k, bool reverse = false) {
    if (!reverse) {
      std::partial_sort(this->begin(), this->begin() + k, this->end());
    } else {
      std::partial_sort(this->begin(), this->begin() + k, this->end(),
                        std::greater<T>());
    }
    return *this;
  }

  Vector<T> &sort() {
    std::sort(this->begin(), this->end());
    return *this;
  }

  template <typename Function> Vector<T> &sort(Function func) {
    std::sort(this->begin(), this->end(), func);
    return *this;
  }

  Vector<T> &rsort() {
    std::sort(this->rbegin(), this->rend());
    return *this;
  }

  Vector<int> argsort() const {
    Vector<Tuple<T, int>> v;
    for (int i = 0; i < this->size(); ++i) {
      v.emplace_back((*this)[i], i);
    }
    v.sort();
    auto f = [](const Tuple<T, int> &t) { return t.template get<1>(); };
    return v.transform(f);
  }

  Vector<T> &nth_element(int n, bool reverse = false) {
    if (!reverse) {
      std::nth_element(this->begin(), this->begin() + n, this->end());
    } else {
      std::nth_element(this->begin(), this->begin() + n, this->end(),
                       std::greater<T>());
    }
    return *this;
  }

  Vector<T> subvector(int a) const {
    return Vector<T>(this->begin(), this->begin() + a);
  }

  Vector<T> subvector(int a, int b) const {
    return Vector<T>(this->begin() + a, this->begin() + b);
  }

  template <typename Function> auto transform(Function func) const {
    Vector<decltype(func(T()))> res;
    std::transform(this->begin(), this->end(), std::back_inserter(res), func);
    return res;
  }

  Vector<T> partial_sum() const {
    Vector<T> res;
    std::partial_sum(this->begin(), this->end(), std::back_inserter(res));
    return res;
  }

  template <typename Function> Vector<T> partial_sum(Function func) const {
    Vector<T> res;
    std::partial_sum(this->begin(), this->end(), std::back_inserter(res), func);
    return res;
  }

  Vector<T> &reverse() {
    std::reverse(this->begin(), this->end());
    return *this;
  }

  template <typename Function> int count_if(Function func) const {
    return std::count_if(this->begin(), this->end(), func);
  }

  Vector<T> adjacent_difference() const {
    Vector<T> res;
    std::adjacent_difference(this->begin(), this->end(),
                             std::back_inserter(res));
    return res;
  }

  T lower_bound(T t) const {
    return std::lower_bound(this->begin(), this->end(), t) - this->begin();
  }

  T upper_bound(T t) const {
    return std::upper_bound(this->begin(), this->end(), t) - this->begin();
  }

  T accumulate() const {
    return std::accumulate(this->begin(), this->end(), T());
  }

  template <typename S, typename Function>
  S accumulate(S n, Function func) const {
    return std::accumulate(this->begin(), this->end(), n, func);
  }

  template <typename Int> static Vector<T> makeVector(Int n) {
    return Vector<T>(n);
  }

  template <typename Int> static Vector<T> makeVector(Input &in, Int n) {
    return Vector<T>(n, in);
  }

  template <typename Int, typename... Ints>
  static auto makeVector(Input &in, Int n, Ints... ints) {
    Vector<decltype(makeVector(in, ints...))> res;
    for (int i = 0; i < n; ++i) {
      res.emplace_back(makeVector(in, ints...));
    }
    return res;
  }

  template <typename Int, typename... Ints>
  static auto makeVector(Int n, Ints... ints) {
    Vector<decltype(makeVector(ints...))> res;
    for (int i = 0; i < n; ++i) {
      res.emplace_back(makeVector(ints...));
    }
    return res;
  }

  Vector<T> &unique() {
    this->erase(std::unique(this->begin(), this->end()), this->end());
    return *this;
  }

  bool next_permutation() {
    return std::next_permutation(this->begin(), this->end());
  }

  Vector<T> &rotate(int n) {
    std::rotate(this->begin(), this->begin() + n, this->end());
    return *this;
  }

  Map<T, int> countAll() const {
    Map<T, int> res;
    for (const auto &i : *this) {
      ++res[i];
    }
    return res;
  }

  T matmul(const T &a) const {
    return this->transform([&](const T &i) { return i.inner_product(a); });
  }
};

template <typename T> Vector<T> iota(int n, T m = 0) {
  Vector<T> v(n);
  std::iota(v.begin(), v.end(), m);
  return v;
}

template <typename T, typename S> void read(Vector<T> &t, Vector<S> &s) {
  for (int i = 0; i < t.size(); ++i) {
    t[i] = T(in);
    s[i] = S(in);
  }
}

template <typename T, typename S, typename U>
void read(Vector<T> &t, Vector<S> &s, Vector<U> &u) {
  for (int i = 0; i < t.size(); ++i) {
    t[i] = T(in);
    s[i] = S(in);
    u[i] = U(in);
  }
}

template <typename T> Vector<T> operator+(const T &a, const Vector<T> &b) {
  return b + a;
}

template <typename T> Vector<T> operator-(const T &a, const Vector<T> &b) {
  return -b + a;
}

template <typename T> Vector<T> operator*(const T &a, const Vector<T> &b) {
  return b * a;
}

/************/
/* math.hpp */
/************/

#include <cmath>

template <typename T = double> constexpr T pi() { return acos(T(-1)); }

template <typename T> T gcd(T t) { return abs(t); }

template <typename T, typename... S> T gcd(T a, S... s) {
  a = abs(a);
  auto b = gcd(s...);
  if (a == 0 || b == 0) {
    return max(a, b);
  }
  int fa = least_bit_fast(a);
  int fb = least_bit_fast(b);
  a >>= fa;
  b >>= fb;
  while (a != b) {
    auto &c = a > b ? a : b;
    c = abs(a - b);
    c >>= least_bit_fast(c);
  }
  return a << min(fa, fb);
}

template <typename T> T gcd(const Vector<T> &v) {
  T g = abs(v[0]);
  for (int i = 1; i < int(v.size()); ++i) {
    g = gcd(g, v[i]);
  }
  return g;
}

template <typename T> T lcm(T t) { return abs(t); }

template <typename T, typename... S> T lcm(T t, S... s) {
  T l = lcm(s...);
  return abs(t) / gcd(t, l) * l;
}

template <typename T> T lcm(const Vector<T> &v) {
  T l = abs(v[0]);
  for (int i = 1; i < int(v.size()); ++i) {
    l = lcm(l, v[i]);
  }
  return l;
}

template <typename T> T floor(T a, T b) {
  auto d = std::div(a, b);
  return d.quot - (d.rem && (a < 0) != (b < 0) ? 1 : 0);
}

template <typename T> T ceil(T a, T b) {
  auto d = std::div(a, b);
  return d.quot + (d.rem && (a > 0) == (b > 0) ? 1 : 0);
}

template <typename T> T round(T a) { return std::round(a); }

template <typename T> T round(T a, T b) { return floor(a + b / 2, b); }

template <typename T> T mod(T a, T b) {
  T c = a % b;
  return c < 0 ? c + abs(b) : c;
}

template <typename T> T factorial(T n) {
  return n <= 1 ? 1 : factorial(n - 1) * n;
}

template <typename T> Vector<T> factorial_vector(int n) {
  Vector<T> v(n + 1, 1);
  for (int i = 1; i <= n; ++i) {
    v[i] = v[i - 1] * i;
  }
  return v;
}

template <typename T> T square(T n) { return n * n; }

template <typename T> T cube(T n) { return n * n * n; }

template <typename T> T norm(T x1, T y1, T x2, T y2) {
  return square(x1 - x2) + square(y1 - y2);
}

template <typename T> bool isSquare(T n) { return square(T(sqrt(n))) == n; }

template <typename T> T clamp(T v, T l, T u) {
  return v < l ? l : v > u ? u : v;
}

template <typename T> auto hypot(T a, T b) { return std::hypot(a, b); }

template <typename T> auto pow(T a, T b) { return std::pow(a, b); }

template <typename T> auto log10(T a) { return std::log10(a); }

/*********************/
/* geometry/real.hpp */
/*********************/

class Real : public Arithmetic<Real>, public Ordered<Real> {
private:
  long double val;

public:
  static long double EPS;

  Real() {}

  Real(long double val) : val(val) {}

  Real operator-() const { return -val; }

  template <typename T> Real &operator+=(const T &r) {
    val += static_cast<long double>(r);
    return *this;
  }

  template <typename T> Real &operator-=(const T &r) {
    val -= static_cast<long double>(r);
    return *this;
  }

  template <typename T> Real &operator*=(const T &r) {
    val *= static_cast<long double>(r);
    return *this;
  }

  template <typename T> Real &operator/=(const T &r) {
    val /= static_cast<long double>(r);
    return *this;
  }

  template <typename T> Real operator-(const T &v) const {
    return Real(*this) -= v;
  }

  template <typename T> Real operator*(const T &v) const {
    return Real(*this) *= v;
  }

  template <typename T> Real operator/(const T &v) const {
    return Real(*this) /= v;
  }

  template <typename T> bool operator<(const T r) const {
    return val < static_cast<long double>(r) - EPS;
  }

  Real abs() const { return std::abs(val); }

  Real sqrt() const { return std::sqrt(val); }

  Real floor() const { return std::floor(val); }

  operator long double() const { return val; }
};

long double Real::EPS = 1e-10;

std::ostream &operator<<(std::ostream &os, const Real &a) {
  os << static_cast<long double>(a);
  return os;
}

/**********************/
/* geometry/point.hpp */
/**********************/

class Point : public Arithmetic<Point>, public Ordered<Point> {
public:
  Real x, y;

  Point() {}

  Point(const Real &x) : x(x), y(0) {}

  Point(const Real &x, const Real &y) : x(x), y(y) {}

  Point(Input &in) : x(in), y(in) {}

  Point &operator+=(const Point &p) {
    x += p.x;
    y += p.y;
    return *this;
  }

  Point &operator-=(const Point &p) {
    x -= p.x;
    y -= p.y;
    return *this;
  }

  Point &operator*=(const Point &p) {
    Real xx = x * p.x - y * p.y;
    y = x * p.y + y * p.x;
    x = xx;
    return *this;
  }

  Point &operator*=(const Real &r) {
    x *= r;
    y *= r;
    return *this;
  }

  Point &operator/=(const Point &p) {
    Real nrm = p.norm();
    Real xx = (x * p.x + y * p.y) / nrm;
    y = (y * p.x - x * p.y) / nrm;
    x = xx;
    return *this;
  }

  Point &operator/=(const Real &r) {
    x /= r;
    y /= r;
    return *this;
  }

  bool operator<(const Point &p) const {
    return (x == p.x) ? y < p.y : x < p.x;
  }

  Real norm() const { return x * x + y * y; }

  Real abs() const { return norm().sqrt(); }

  Point conj() const { return Point(x, -y); }
};

Point operator*(const Real &real, const Point &point) { return point * real; }

Point operator/(const Real &real, const Point &point) { return point / real; }

std::ostream &operator<<(std::ostream &os, const Point &point) {
  os << point.x << " " << point.y;
  return os;
}

/*********************/
/* geometry/line.hpp */
/*********************/

class Line {
public:
  Point a, b;

  Line() {}

  Line(const Point &a, const Point &b) : a(a), b(b) {}

  Line(Input &in) : a(in), b(in) {}

  bool operator==(const Line &line) const {
    return ((line.vec() / vec()).y == 0) && (((line.a - a) / vec()).y == 0);
  }

  Point vec() const { return b - a; }
};

std::ostream &operator<<(std::ostream &os, const Line &line) {
  os << line.a << " " << line.b;
  return os;
}

/****************************/
/* geometry/cross_point.hpp */
/****************************/

Point crossPoint(const Line &line1, const Line &line2) {
  return line1.a + line1.vec() * ((line2.a - line1.a) / line2.vec()).y /
                       (line1.vec() / line2.vec()).y;
}
/************************/
/* geometry/segment.hpp */
/************************/

class Segment : public Line, public Ordered<Segment> {
public:
  Segment() {}

  Segment(const Point &a, const Point &b) : Line(a, b) {}

  Segment(Input &in) : Line(in) {}

  bool operator<(const Segment &segment) const {
    return a == segment.a ? b < segment.b : a < segment.a;
  }

  Real area() const {
    return (this->a.x * this->b.y - this->a.y * this->b.x) / 2;
  }
};

/********************/
/* geometry/ccw.hpp */
/********************/

enum CCW { LEFT = 1, RIGHT = 2, BACK = 4, FRONT = 8, ON = 16 };

int ccw(const Point &a, const Point &b, const Point &c) {
  Point p = (c - a) / (b - a);
  if (p.y > 0) {
    return LEFT;
  }
  if (p.y < 0) {
    return RIGHT;
  }
  if (p.x < 0) {
    return BACK;
  }
  if (p.x > 1) {
    return FRONT;
  }
  return ON;
}

int ccw(const Segment &segment, const Point &point) {
  return ccw(segment.a, segment.b, point);
}

int ccw(const Line &line, const Point &point) {
  int res = ccw(line.a, line.b, point);
  if (res == BACK || res == FRONT) {
    res = ON;
  }
  return res;
}

/**************************/
/* geometry/intersect.hpp */
/**************************/

template <bool strict = false>
bool intersect(const Line &line1, const Line &line2) {
  if (strict) {
    return (line1.vec() / line2.vec()).y != 0;
  }
  return ((line1.vec() / line2.vec()).y != 0) || (line1 == line2);
}

template <bool strict = false>
bool intersect(const Line &line, const Segment &segment) {
  Point p1 = (segment.a - line.a) / line.vec();
  Point p2 = (segment.b - line.a) / line.vec();
  if (strict) {
    return p1.y * p2.y < 0;
  }
  return p1.y * p2.y <= 0;
}

template <bool strict = false>
bool intersect(const Segment &segment, const Line &line) {
  return intersect(line, segment);
}

template <bool strict = false>
bool intersect(const Segment &segment1, const Segment &segment2) {
  int ccw1 = ccw(segment1, segment2.a) | ccw(segment1, segment2.b);
  int ccw2 = ccw(segment2, segment1.a) | ccw(segment2, segment1.b);
  if (strict) {
    return (ccw1 & ccw2) == (LEFT | RIGHT);
  }
  return ((ccw1 & ccw2) == (LEFT | RIGHT)) || ((ccw1 | ccw2) & ON);
}
/************************/
/* geometry/polygon.hpp */
/************************/

class Polygon : public Vector<Point> {
public:
  Polygon() {}

  Polygon(int n) : Vector<Point>(n) {}

  Polygon(int n, Input &in) : Vector<Point>(n, in) {}

  Polygon(const std::initializer_list<Point> &p) : Vector<Point>(p) {}

  Polygon(const Vector<Point> &p) : Vector<Point>(p) {}

  Vector<Segment> getSides() const {
    if (size() <= 1) {
      return {};
    }
    Vector<Segment> res;
    Point pre = back();
    for (const auto &point : *this) {
      res.emplace_back(pre, point);
      pre = point;
    }
    return res;
  }

  Vector<Vector<Point>> getCorners() const {
    if (size() <= 2) {
      return {};
    }
    Vector<Vector<Point>> res;
    Point pre1 = *(end() - 2), pre2 = back();
    for (const auto &point : *this) {
      res.emplace_back(Vector<Point>({pre1, pre2, point}));
      pre1 = pre2;
      pre2 = point;
    }
    return res;
  }

  Point &operator[](int i) { return vector::operator[](mod(i, size())); }

  const Point &operator[](int i) const {
    return vector::operator[](mod(i, size()));
  }

  template <bool strict = false> bool cover(const Point &point) const {
    bool res = false;
    for (auto &side : getSides()) {
      if (ccw(side, point) == ON) {
        return strict ? false : true;
      }
      if (side.a.y > side.b.y) {
        std::swap(side.a, side.b);
      }
      if (side.a.y <= point.y && point.y < side.b.y &&
          ((side.b - point) / (side.a - point)).y > 0) {
        res = !res;
      }
    }
    return res;
  }

  Real area() const {
    auto side = this->getSides();
    auto f = [](const Real &r, const Segment &segment) {
      return r + segment.area();
    };
    return side.accumulate(0, f);
  }
};

/*******************************/
/* geometry/convex_polygon.hpp */
/*******************************/

template <bool strict = true> class ConvexPolygon : public Polygon {
public:
  ConvexPolygon() {}

  ConvexPolygon(int n) : Polygon(n) {}

  ConvexPolygon(Vector<Point> points) {
    int flag = ~(strict ? LEFT : LEFT | FRONT);
    points.sort();
    for (int i = 0; i < points.size(); emplace_back(points[i++])) {
      while (size() > 1 && ccw(*(end() - 2), back(), points[i]) & flag) {
        pop_back();
      }
    }
    for (int i = points.size() - 2, r = size(); i >= 0;
         emplace_back(points[i--])) {
      while (size() > r && ccw(*(end() - 2), back(), points[i]) & flag) {
        pop_back();
      }
    }
    pop_back();
  }

  Real diameter() {
    auto sides = getSides();
    int i = min_element(sides.begin(), sides.end()) - sides.begin();
    int j = max_element(sides.begin(), sides.end()) - sides.begin();
    sides.insert(sides.end(), sides.begin(), sides.end());
    Real res = 0;
    for (int k = 0; k < 2 * size(); ++k) {
      if ((sides[i].vec() / sides[j].vec()).y >= 0) {
        ++i;
      } else {
        ++j;
      }
      chmax(res, (sides[i].a - sides[j].a).abs());
    }
    return res;
  }

  ConvexPolygon cut(const Line &line) {
    ConvexPolygon res;
    auto sides = getSides();
    for (const auto &side : sides) {
      if (ccw(line, side.a) != RIGHT) {
        res.push_back(side.a);
      }
      if (intersect<true>(line, side)) {
        res.push_back(crossPoint(line, side));
      }
    }
    return res;
  }
};

/************/
/* main.cpp */
/************/

int main() {
  setBoolName("YES", "NO");
  Vector<Point> p(5, in);
  cout << (ConvexPolygon<>(p).size() == 5) << endl;
}