#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 pi = pair<ll, ll>;
using vi = vector<ll>;
using u32 = unsigned int;
using u64 = unsigned long long;
using i128 = __int128;

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) {
  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 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})); }
};

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 T>
struct Circle {
  Point<T> O;
  T r;
  Circle(Point<T> O, T r) : O(O), r(r) {}
  Circle(T x, T y, T r) : O(Point<T>(x, y)), 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/cross_point.hpp"

// 平行でないことを仮定
template <typename REAL, typename T>
Point<REAL> cross_point(const Line<T> L1, const Line<T> L2) {
  T det = L1.a * L2.b - L1.b * L2.a;
  assert(det != 0);
  REAL x = -REAL(L1.c) * L2.b + REAL(L1.b) * L2.c;
  REAL y = -REAL(L1.a) * L2.c + REAL(L1.c) * L2.a;
  return Point<REAL>(x / det, y / det);
}

// 0: 交点なし
// 1: 一意な交点
// 2：2 つ以上の交点（整数型を利用して厳密にやる）
template <typename T, enable_if_t<is_integral<T>::value, int> = 0>
int count_cross(Segment<T> S1, Segment<T> S2, bool include_ends) {
  Line<T> L1 = S1.to_Line();
  Line<T> L2 = S2.to_Line();
  if (L1.is_parallel(L2)) {
    if (L1.eval(S2.A) != 0) return 0;
    // 4 点とも同一直線上にある
    T a1 = S1.A.x, b1 = S1.B.x;
    T a2 = S2.A.x, b2 = S2.B.x;
    if (a1 == b1) {
      a1 = S1.A.y, b1 = S1.B.y;
      a2 = S2.A.y, b2 = S2.B.y;
    }
    if (a1 > b1) swap(a1, b1);
    if (a2 > b2) swap(a2, b2);
    T a = max(a1, a2);
    T b = min(b1, b2);
    if (a < b) return 2;
    if (a > b) return 0;
    return (include_ends ? 1 : 0);
  }
  // 平行でない場合
  T a1 = L2.eval(S1.A), b1 = L2.eval(S1.B);
  T a2 = L1.eval(S2.A), b2 = L1.eval(S2.B);
  if (a1 > b1) swap(a1, b1);
  if (a2 > b2) swap(a2, b2);
  bool ok1 = 0, ok2 = 0;

  if (include_ends) {
    ok1 = (a1 <= 0) && (0 <= b1);
    ok2 = (a2 <= 0) && (0 <= b2);
  } else {
    ok1 = (a1 < 0) && (0 < b1);
    ok2 = (a2 < 0) && (0 < b2);
  }
  return (ok1 && ok2 ? 1 : 0);
}

// 0 または 1
// real だと誤差によっては間違った答を返す可能性あり。
/*
template <typename T>
int count_cross(Segment<T> S1, Segment<T> S2) {
  Line<T> L1 = S1.to_Line();
  Line<T> L2 = S2.to_Line();
  T a1 = L2.eval(S1.A), b1 = L2.eval(S1.B);
  T a2 = L1.eval(S2.A), b2 = L1.eval(S2.B);
  if (a1 > b1) swap(a1, b1);
  if (a2 > b2) swap(a2, b2);
  bool ok1 = 0, ok2 = 0;
  ok1 = (a1 <= 0) && (0 <= b1);
  ok2 = (a2 <= 0) && (0 <= b2);
  return (ok1 && ok2 ? 1 : 0);
}
*/

// 唯一の交点を持つことを仮定
template <typename REAL, typename T>
Point<REAL> cross_point(Segment<T> S1, Segment<T> S2) {
  Line<T> L1 = S1.to_Line();
  Line<T> L2 = S2.to_Line();
  return cross_point<REAL, T>(L1, L2);
}
#line 2 "library/mod/modint.hpp"

template <int mod>
struct modint {
  int val;
  constexpr modint(ll x = 0) noexcept {
    if (0 <= x && x < mod)
      val = x;
    else {
      x %= mod;
      val = (x < 0 ? x + mod : x);
    }
  }
  bool operator<(const modint &other) const {
    return val < other.val;
  } // To use std::map
  modint &operator+=(const modint &p) {
    if ((val += p.val) >= mod) val -= mod;
    return *this;
  }
  modint &operator-=(const modint &p) {
    if ((val += mod - p.val) >= mod) val -= mod;
    return *this;
  }
  modint &operator*=(const modint &p) {
    val = (int)(1LL * val * p.val % mod);
    return *this;
  }
  modint &operator/=(const modint &p) {
    *this *= p.inverse();
    return *this;
  }
  modint operator-() const { return modint(-val); }
  modint operator+(const modint &p) const { return modint(*this) += p; }
  modint operator-(const modint &p) const { return modint(*this) -= p; }
  modint operator*(const modint &p) const { return modint(*this) *= p; }
  modint operator/(const modint &p) const { return modint(*this) /= p; }
  bool operator==(const modint &p) const { return val == p.val; }
  bool operator!=(const modint &p) const { return val != p.val; }
  modint inverse() const {
    int a = val, b = mod, u = 1, v = 0, t;
    while (b > 0) {
      t = a / b;
      swap(a -= t * b, b), swap(u -= t * v, v);
    }
    return modint(u);
  }
  modint pow(int64_t n) const {
    modint ret(1), mul(val);
    while (n > 0) {
      if (n & 1) ret *= mul;
      mul *= mul;
      n >>= 1;
    }
    return ret;
  }
#ifdef FASTIO
  void write() { fastio::printer.write(val); }
  void read() {
    ll x;
    fastio::scanner.read(x);
    if (x < 0 || x >= mod) x %= mod;
    if (x < 0) x += mod;
    val += x;
  }
#endif
  static constexpr int get_mod() { return mod; }
};

struct ArbitraryModInt {
  static constexpr bool is_modint = true;
  int val;
  ArbitraryModInt() : val(0) {}
  ArbitraryModInt(int64_t y)
      : val(y >= 0 ? y % get_mod()
                   : (get_mod() - (-y) % get_mod()) % get_mod()) {}
  bool operator<(const ArbitraryModInt &other) const {
    return val < other.val;
  } // To use std::map<ArbitraryModInt, T>
  static int &get_mod() {
    static int mod = 0;
    return mod;
  }
  static void set_mod(int md) { get_mod() = md; }
  ArbitraryModInt &operator+=(const ArbitraryModInt &p) {
    if ((val += p.val) >= get_mod()) val -= get_mod();
    return *this;
  }
  ArbitraryModInt &operator-=(const ArbitraryModInt &p) {
    if ((val += get_mod() - p.val) >= get_mod()) val -= get_mod();
    return *this;
  }
  ArbitraryModInt &operator*=(const ArbitraryModInt &p) {
    long long a = (long long)val * p.val;
    int xh = (int)(a >> 32), xl = (int)a, d, m;
    asm("divl %4; \n\t" : "=a"(d), "=d"(m) : "d"(xh), "a"(xl), "r"(get_mod()));
    val = m;
    return *this;
  }
  ArbitraryModInt &operator/=(const ArbitraryModInt &p) {
    *this *= p.inverse();
    return *this;
  }
  ArbitraryModInt operator-() const { return ArbitraryModInt(get_mod() - val); }
  ArbitraryModInt operator+(const ArbitraryModInt &p) const {
    return ArbitraryModInt(*this) += p;
  }
  ArbitraryModInt operator-(const ArbitraryModInt &p) const {
    return ArbitraryModInt(*this) -= p;
  }
  ArbitraryModInt operator*(const ArbitraryModInt &p) const {
    return ArbitraryModInt(*this) *= p;
  }
  ArbitraryModInt operator/(const ArbitraryModInt &p) const {
    return ArbitraryModInt(*this) /= p;
  }
  bool operator==(const ArbitraryModInt &p) const { return val == p.val; }
  bool operator!=(const ArbitraryModInt &p) const { return val != p.val; }
  ArbitraryModInt inverse() const {
    int a = val, b = get_mod(), u = 1, v = 0, t;
    while (b > 0) {
      t = a / b;
      swap(a -= t * b, b), swap(u -= t * v, v);
    }
    return ArbitraryModInt(u);
  }
  ArbitraryModInt pow(int64_t n) const {
    ArbitraryModInt ret(1), mul(val);
    while (n > 0) {
      if (n & 1) ret *= mul;
      mul *= mul;
      n >>= 1;
    }
    return ret;
  }
#ifdef FASTIO
  void write() { fastio::printer.write(val); }
  void read() { fastio::scanner.read(val); }
#endif
};

template <typename mint>
mint inv(int n) {
  static const int mod = mint::get_mod();
  static vector<mint> dat = {0, 1};
  assert(0 <= n);
  if (n >= mod) n %= mod;
  while (int(dat.size()) <= n) {
    int k = dat.size();
    auto q = (mod + k - 1) / k;
    int r = k * q - mod;
    dat.emplace_back(dat[r] * mint(q));
  }
  return dat[n];
}

template <typename mint>
mint fact(int n) {
  static const int mod = mint::get_mod();
  static vector<mint> dat = {1, 1};
  assert(0 <= n);
  if (n >= mod) return 0;
  while (int(dat.size()) <= n) {
    int k = dat.size();
    dat.emplace_back(dat[k - 1] * mint(k));
  }
  return dat[n];
}

template <typename mint>
mint fact_inv(int n) {
  static const int mod = mint::get_mod();
  static vector<mint> dat = {1, 1};
  assert(-1 <= n && n < mod);
  if (n == -1) return mint(0);
  while (int(dat.size()) <= n) {
    int k = dat.size();
    dat.emplace_back(dat[k - 1] * inv<mint>(k));
  }
  return dat[n];
}

template <class mint, class... Ts>
mint fact_invs(Ts... xs) {
  return (mint(1) * ... * fact_inv<mint>(xs));
}

template <typename mint, class Head, class... Tail>
mint multinomial(Head &&head, Tail &&... tail) {
  return fact<mint>(head) * fact_invs<mint>(std::forward<Tail>(tail)...);
}

template <typename mint>
mint C_dense(int n, int k) {
  static vvc<mint> C;
  static int H = 0, W = 0;

  auto calc = [&](int i, int j) -> mint {
    if (i == 0) return (j == 0 ? mint(1) : mint(0));
    return C[i - 1][j] + (j ? C[i - 1][j - 1] : 0);
  };

  if (W <= k) {
    FOR(i, H) {
      C[i].resize(k + 1);
      FOR(j, W, k + 1) { C[i][j] = calc(i, j); }
    }
    W = k + 1;
  }
  if (H <= n) {
    C.resize(n + 1);
    FOR(i, H, n + 1) {
      C[i].resize(W);
      FOR(j, W) { C[i][j] = calc(i, j); }
    }
    H = n + 1;
  }
  return C[n][k];
}

template <typename mint, bool large = false, bool dense = false>
mint C(ll n, ll k) {
  assert(n >= 0);
  if (k < 0 || n < k) return 0;
  if (dense) return C_dense<mint>(n, k);
  if (!large) return fact<mint>(n) * fact_inv<mint>(k) * fact_inv<mint>(n - k);
  k = min(k, n - k);
  mint x(1);
  FOR(i, k) { x *= mint(n - i); }
  x *= fact_inv<mint>(k);
  return x;
}

template <typename mint, bool large = false>
mint C_inv(ll n, ll k) {
  assert(n >= 0);
  assert(0 <= k && k <= n);
  if (!large) return fact_inv<mint>(n) * fact<mint>(k) * fact<mint>(n - k);
  return mint(1) / C<mint, 1>(n, k);
}

// [x^d] (1-x) ^ {-n} の計算
template <typename mint, bool large = false, bool dense = false>
mint C_negative(ll n, ll d) {
  assert(n >= 0);
  if (d < 0) return mint(0);
  if (n == 0) { return (d == 0 ? mint(1) : mint(0)); }
  return C<mint, large, dense>(n + d - 1, d);
}

using modint107 = modint<1000000007>;
using modint998 = modint<998244353>;
using amint = ArbitraryModInt;
#line 2 "library/nt/primetable.hpp"

template <typename T = long long>
vc<T> primetable(int LIM) {
  ++LIM;
  const int S = 32768;
  static int done = 2;
  static vc<T> primes = {2}, sieve(S + 1);

  if (done < LIM) {
    done = LIM;

    primes = {2}, sieve.assign(S + 1, 0);
    const int R = LIM / 2;
    primes.reserve(int(LIM / log(LIM) * 1.1));
    vc<pair<int, int>> cp;
    for (int i = 3; i <= S; i += 2) {
      if (!sieve[i]) {
        cp.eb(i, i * i / 2);
        for (int j = i * i; j <= S; j += 2 * i) sieve[j] = 1;
      }
    }
    for (int L = 1; L <= R; L += S) {
      array<bool, S> block{};
      for (auto& [p, idx]: cp)
        for (int i = idx; i < S + L; idx = (i += p)) block[i - L] = 1;
      FOR(i, min(S, R - L)) if (!block[i]) primes.eb((L + i) * 2 + 1);
    }
  }
  int k = LB(primes, LIM + 1);
  return {primes.begin(), primes.begin() + k};
}
#line 3 "library/mod/powertable.hpp"

// a^0, ..., a^N
template <typename mint>
vc<mint> powertable_1(mint a, ll N) {
  // table of a^i
  vc<mint> f(N + 1, 1);
  FOR(i, N) f[i + 1] = a * f[i];
  return f;
}

// 0^e, ..., N^e
template <typename mint>
vc<mint> powertable_2(ll e, ll N) {
  auto primes = primetable(N);
  vc<mint> f(N + 1, 1);
  f[0] = mint(0).pow(e);
  for (auto&& p: primes) {
    if (p > N) break;
    mint xp = mint(p).pow(e);
    ll pp = p;
    while (pp <= N) {
      ll i = pp;
      while (i <= N) {
        f[i] *= xp;
        i += pp;
      }
      pp *= p;
    }
  }
  return f;
}
#line 2 "library/geo/angle_sort.hpp"

#line 4 "library/geo/angle_sort.hpp"

// 偏角ソートに対する argsort
template <typename T>
vector<int> angle_argsort(vector<Point<T>>& P) {
  vector<int> lower, origin, upper;
  const Point<T> O = {0, 0};
  FOR(i, len(P)) {
    if (P[i] == O) origin.eb(i);
    elif ((P[i].y < 0) || (P[i].y == 0 && P[i].x > 0)) lower.eb(i);
    else upper.eb(i);
  }
  sort(all(lower), [&](auto& i, auto& j) { return P[i].det(P[j]) > 0; });
  sort(all(upper), [&](auto& i, auto& j) { return P[i].det(P[j]) > 0; });
  auto& I = lower;
  I.insert(I.end(), all(origin));
  I.insert(I.end(), all(upper));
  return I;
}

// 偏角ソートに対する argsort
template <typename T>
vector<int> angle_argsort(vector<pair<T, T>>& P) {
  vc<Point<T>> tmp(len(P));
  FOR(i, len(P)) tmp[i] = Point<T>(P[i]);
  return angle_argsort<T>(tmp);
}

// inplace に偏角ソートする
// index が欲しい場合は angle_argsort
template <typename T>
void angle_sort(vector<Point<T>>& P) {
  auto I = angle_argsort<T>(P);
  P = rearrange(P, I);
}

// inplace に偏角ソートする
// index が欲しい場合は angle_argsort
template <typename T>
void angle_sort(vector<pair<T, T>>& P) {
  auto I = angle_argsort<T>(P);
  P = rearrange(P, I);
}
#line 8 "main.cpp"

using P = Point<ll>;
using mint = modint998;

/*
凸多角形を作る dp。
・正の角まわる移動だけを考える。間の点があれば 2^n かける。
・「2 角形」はあとで処理する
*/

void solve() {
  LL(N);
  VEC(P, dat, N);
  vv(bool, CAN, N, N);
  vv(int, CNT, N, N);

  auto PREV = [&](int i) -> int { return (i == 0 ? N : i) - 1; };
  auto NXT = [&](int i) -> int { return (i == N - 1 ? 0 : i + 1); };

  FOR(a, N) FOR(b, N) {
    if (a == b) continue;
    P A = dat[a], B = dat[b];
    Segment<ll> AB(A, B);
    Line<ll> L = AB.to_Line();

    vc<bool> ON(N);
    FOR(c, N) {
      if (AB.contain(dat[c])) ON[c] = 1;
    }
    CNT[a][b] = SUM<int>(ON) - 2;
    bool ok = 1;

    // 線分との非自明な交差
    FOR(c, N) {
      int d = NXT(c);
      if (ON[c] || ON[d]) continue;
      P C = dat[c], D = dat[d];
      Segment<ll> CD(C, D);
      if (count_cross(AB, CD, 1)) { ok = 0; }
    }
    // ちょうど乗ってる 1 点の前後
    FOR(c, N) {
      if (a == c || b == c) continue;
      if (!ON[c]) continue;
      int x = PREV(c);
      int y = NXT(c);
      if (ON[x] || ON[y]) continue;
      ll xx = L.eval(dat[x]);
      ll yy = L.eval(dat[y]);
      if (xx > 0 && yy < 0) ok = 0;
      if (xx < 0 && yy > 0) ok = 0;
    }
    // 乗ってる 2 点の前後
    FOR(c, N) {
      int d = NXT(c);
      if (a == c || b == c || a == d || b == d) continue;
      if (!ON[c] || !ON[d]) continue;
      int x = (c == 0 ? N : c) - 1;
      int y = (d + 1 == N ? 0 : d + 1);
      ll xx = L.eval(dat[x]);
      ll yy = L.eval(dat[y]);
      if (xx > 0 && yy < 0) ok = 0;
      if (xx < 0 && yy > 0) ok = 0;
    }
    if (!ok) continue;
    // 交わっていないことまでチェックした。
    // 外側を進んでいないかどうかを確認しないといけない
    // A の近傍だけ見ればよい
    int c = PREV(a), d = NXT(a);
    if (c == b || d == b) {
      // ok
    } else {
      P C = dat[c], D = dat[d];
      bool left = (A - C).det(D - A) > 0;
      if (left) {
        ok &= (A - C).det(B - A) >= 0 && (D - A).det(B - A) >= 0;
      } else {
        ok &= (A - C).det(B - A) >= 0 || (D - A).det(B - A) >= 0;
      }
    }
    CAN[a][b] = ok;
  }

  /*
  FOR(a, N) print(CAN[a]);
  print();
  FOR(a, N) print(CNT[a]);
  print();
  */

  FOR(a, N) FOR(b, N) assert(CAN[a][b] == CAN[b][a]);
  FOR(a, N) FOR(b, N) assert(CNT[a][b] == CNT[b][a]);

  mint ANS = 0;
  vc<mint> POW = powertable_1<mint>(2, N);
  // 凸包が線分
  FOR(a, N) FOR(b, a) {
    if (CAN[a][b]) ANS += POW[CNT[a][b]];
  }

  // angle sort
  vc<pi> edge;
  vc<P> dir;
  FOR(a, N) FOR(b, N) {
    if (CAN[a][b]) {
      edge.eb(a, b);
      dir.eb(dat[b] - dat[a]);
    }
  }
  auto I = angle_argsort(dir);
  edge = rearrange(edge, I);
  dir = rearrange(dir, I);

  FOR(s, N) {
    // s スタート
    // [0][v]：1 回だけ進んだ
    // [1][v]：2 回以上進んだ
    vv(mint, dp, 2, N);
    // 同じ向きの線分での遷移をまとめて更新するようにする
    auto SAME = [&](int i, int j) -> bool {
      return (dir[i].det(dir[j]) == 0 && dir[i].dot(dir[j]) > 0);
    };
    int L = 0;
    while (L < len(edge)) {
      int R = L;
      while (R < len(edge) && SAME(L, R)) ++R;
      vc<tuple<int, int, mint>> upd;
      FOR(e, L, R) {
        auto [frm, to] = edge[e];
        mint cf = POW[CNT[frm][to]];
        if (frm == s) {
          upd.eb(0, to, cf);
        } else {
          if (to != s) upd.eb(1, to, dp[0][frm] * cf);
          upd.eb(1, to, dp[1][frm] * cf);
        }
      }
      L = R;
      for (auto&& [a, b, c]: upd) dp[a][b] += c;
    }
    ANS += dp[1][s];
  }
  print(ANS);
}

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