description: This is the hard version of the problem. The only difference from the easy version is that in this version the coordinates can be both odd and even.
  
  There are n fence-posts at distinct coordinates on a plane. It is guaranteed that no three fence posts lie on the same line.
  
  There are an infinite number of cows on the plane, one at every point with integer coordinates.
  
  Gregor is a member of the Illuminati, and wants to build a triangular fence, connecting 3 distinct existing fence posts. A cow strictly inside the fence is said to be enclosed. If there are an odd number of enclosed cows and the area of the fence is an integer, the fence is said to be interesting.
  
  Find the number of interesting fences.
  
  Input
  
  The first line contains the integer n (3 ≤ n ≤ 6000), the number of fence posts which Gregor can choose to form the vertices of a fence.
  
  Each of the next n line contains two integers x and y (0 ≤ x,y ≤ 10^7, where (x,y) is the coordinate of a fence post. All fence posts lie at distinct coordinates. No three fence posts are on the same line.
  
  Output
  
  Print a single integer, the number of interesting fences. Two fences are considered different if they are constructed with a different set of three fence posts.
  
  Examples
  
  Input
  
  
  3
  0 0
  2 0
  0 4
  
  
  Output
  
  
  1
  
  
  Input
  
  
  4
  1 8
  0 6
  5 2
  5 6
  
  
  Output
  
  
  1
  
  
  Input
  
  
  10
  170 59
  129 54
  5 98
  129 37
  58 193
  154 58
  24 3
  13 138
  136 144
  174 150
  
  
  Output
  
  
  29
  
  Note
  
  In the first example, there is only 1 fence. That fence is interesting since its area is 4 and there is 1 enclosed cow, marked in red.
  
  <image>
  
  In the second example, there are 4 possible fences. Only one of them is interesting however. That fence has an area of 8 and 5 enclosed cows.
  
  <image>
solution: #include <bits/stdc++.h>
  class Scanner {
   private:
    static const int BUFFER_SIZE = 1 << 18;
    char buff[BUFFER_SIZE];
    char *buffPos, *buffLim;
    FILE* file;
  
   public:
    Scanner(FILE* file) {
      this->file = file;
      buffLim = buff + fread(buff, 1, BUFFER_SIZE, file);
      buffPos = buff;
    }
  
   private:
    inline void flushBuff() {
      buffLim = buff + fread(buff, 1, BUFFER_SIZE, file);
      if (buffLim == buff) {
        *buffLim++ = '\n';
      }
      buffPos = buff;
    }
    inline bool isWS(char t) { return t <= ' '; }
    inline bool isDig(char t) { return t >= '0' && t <= '9'; }
    inline void nextPos() {
      buffPos++;
      if (buffPos == buffLim) {
        flushBuff();
      }
    }
  
   public:
    inline char getchar() {
      char ch = *buffPos;
      nextPos();
      return ch;
    }
    inline void next(char* s) {
      while (isWS(*buffPos)) {
        nextPos();
      }
      while (!isWS(*buffPos)) {
        *s = *buffPos;
        s++;
        nextPos();
      }
      *s = '\0';
    }
    inline void nextLine(char* s) {
      while (*buffPos != '\n') {
        nextPos();
      }
      if (*buffPos == '\n') {
        nextPos();
      }
      while (*buffPos != '\n') {
        *s++ = *buffPos;
        nextPos();
      }
      *s = '\0';
    }
    inline int nextInt() {
      while (!isDig(*buffPos) && *buffPos != '-') {
        nextPos();
      }
      int sign = (*buffPos == '-') ? nextPos(), -1 : 1;
      int res = 0;
      while (isDig(*buffPos)) {
        res = res * 10 + *buffPos - '0';
        nextPos();
      }
      return res * sign;
    }
    inline long long nextLong() {
      while (!isDig(*buffPos) && *buffPos != '-') {
        nextPos();
      }
      long long sign = (*buffPos == '-') ? nextPos(), -1 : 1;
      long long res = 0;
      while (isDig(*buffPos)) {
        res = res * 10 + *buffPos - '0';
        nextPos();
      }
      return res * sign;
    }
    inline int n() {
      while (*buffPos < '0' || *buffPos > '9') {
        buffPos++;
        if (buffPos == buffLim) {
          flushBuff();
        }
      }
      int res = 0;
      while (*buffPos >= '0' && *buffPos <= '9') {
        res = res * 10 + (*buffPos - '0');
        buffPos++;
        if (buffPos == buffLim) {
          flushBuff();
        }
      }
      return res;
    }
    inline long long nl() {
      while (*buffPos < '0' || *buffPos > '9') {
        buffPos++;
        if (buffPos == buffLim) {
          flushBuff();
        }
      }
      long long res = 0;
      while (*buffPos >= '0' && *buffPos <= '9') {
        res = res * 10 + (*buffPos - '0');
        buffPos++;
        if (buffPos == buffLim) {
          flushBuff();
        }
      }
      return res;
    }
    inline long long nlm(const int MOD) {
      while (*buffPos < '0' || *buffPos > '9') {
        buffPos++;
        if (buffPos == buffLim) {
          flushBuff();
        }
      }
      long long res = 0;
      while (*buffPos >= '0' && *buffPos <= '9') {
        res = (res * 10 + (*buffPos - '0')) % MOD;
        buffPos++;
        if (buffPos == buffLim) {
          flushBuff();
        }
      }
      return res;
    }
    inline double nextDouble() {
      while (isWS(*buffPos)) {
        nextPos();
      }
      int sign = (*buffPos == '-') ? nextPos(), -1 : 1;
      double res = 0;
      while (isDig(*buffPos)) {
        res = res * 10 + *buffPos - '0';
        nextPos();
      }
      if (*buffPos == '.') {
        nextPos();
        double ep = 1;
        while (isDig(*buffPos)) {
          ep *= 0.1;
          res += ep * (*buffPos - '0');
          nextPos();
        }
      }
      return sign * res;
    }
    inline char nextChar() {
      while (isWS(*buffPos)) nextPos();
      char res = *buffPos;
      nextPos();
      return res;
    }
    ~Scanner() { fclose(file); }
  };
  Scanner sc(stdin);
  template <class __TyFirst, class __TySecond>
  std::ostream& operator<<(std::ostream& out,
                           const std::pair<__TyFirst, __TySecond>& o);
  static std::string cout_list_sep = " ";
  template <
      class IterateType, typename = decltype(std::declval<IterateType>().begin()),
      typename = decltype(std::declval<IterateType>().end()),
      typename = typename std::enable_if<std::is_convertible<
          typename std::iterator_traits<
              typename IterateType::iterator>::iterator_category,
          std::input_iterator_tag>::value>::type,
      typename = typename std::enable_if<!std::is_base_of<
          std::string, typename std::decay<IterateType>::type>::value>::type,
      typename value_type = typename IterateType::value_type>
  std::ostream& operator<<(std::ostream& out, const IterateType& a) {
    for (const auto& i : a) out << i << cout_list_sep;
    return out;
  }
  template <class __TyFirst, class __TySecond>
  std::ostream& operator<<(std::ostream& out,
                           const std::pair<__TyFirst, __TySecond>& o) {
    out << "(" << o.first << "," << o.second << ")";
    return out;
  }
  template <typename _ForwardIterator>
  void logArray(_ForwardIterator, _ForwardIterator, const char* = 0) {}
  namespace Temps {
  template <typename _IntType>
  class RangeInt {
    static_assert(std::is_integral<_IntType>::value,
                  "RangeInt object must have integral value type");
  
   public:
    class iterator : public std::iterator<std::input_iterator_tag, _IntType,
                                          _IntType, const _IntType*, _IntType> {
      _IntType val, step;
  
     public:
      using typename std::iterator<std::input_iterator_tag, _IntType, _IntType,
                                   const _IntType*, _IntType>::reference;
      explicit constexpr iterator(int val, int step) noexcept
          : val(val), step(step) {}
      constexpr iterator& operator++() noexcept {
        val += step;
        return *this;
      }
      constexpr iterator operator++(int) noexcept {
        iterator ret = *this;
        val += step;
        return ret;
      }
      constexpr bool operator==(const iterator& rhs) const noexcept {
        return val == rhs.val;
      }
      constexpr bool operator!=(const iterator& rhs) const noexcept {
        return val != rhs.val;
      }
      constexpr reference operator*() const { return val; }
    };
    const _IntType _begin, _step, _end;
    explicit constexpr RangeInt(const _IntType end) noexcept
        : _begin(0), _step(1), _end(end > 0 ? end : 0) {}
    explicit constexpr RangeInt(const _IntType begin, const _IntType end) noexcept
        : _begin(begin), _step(1), _end(end > begin ? end : begin) {}
    explicit constexpr RangeInt(const _IntType begin, const _IntType end,
                                const _IntType step)
        : _begin(begin),
          _step(step),
          _end(((step > 0 && end <= begin) || (step < 0 && end >= begin))
                   ? begin
                   : (step > 0
                          ? begin + (end - begin + step - 1) / step * step
                          : begin + (begin - end - step - 1) / (-step) * step)) {}
    constexpr iterator begin() const noexcept { return iterator(_begin, _step); }
    constexpr iterator end() const noexcept { return iterator(_end, _step); }
    constexpr _IntType size() const noexcept { return (_end - _begin) / _step; }
  };
  template <class IntType, typename = typename std::enable_if<
                               std::is_integral<IntType>::value>::type>
  inline constexpr RangeInt<IntType> range(IntType arg1) {
    return RangeInt<IntType>(arg1);
  }
  template <class IntType, typename = typename std::enable_if<
                               std::is_integral<IntType>::value>::type>
  inline constexpr RangeInt<IntType> range(IntType arg1, IntType arg2) {
    return RangeInt<IntType>(arg1, arg2);
  }
  template <class IntType, typename = typename std::enable_if<
                               std::is_integral<IntType>::value>::type>
  inline constexpr RangeInt<IntType> range(IntType arg1, IntType arg2,
                                           IntType arg3) {
    return RangeInt<IntType>(arg1, arg2, arg3);
  }
  }  // namespace Temps
  using Temps::range;
  namespace Temps {
  template <class T>
  inline bool checkMin(T& a, T b) {
    return (b < a ? a = b, 1 : 0);
  }
  template <class T>
  inline bool checkMax(T& a, T b) {
    return (a < b ? a = b, 1 : 0);
  }
  template <class IntType, typename = typename std::enable_if<
                               std::is_integral<IntType>::value>::type>
  IntType gcd(const IntType a, const IntType b) {
    return b == 0 ? a : gcd(b, a % b);
  }
  template <class IntType, typename = typename std::enable_if<
                               std::is_integral<IntType>::value>::type>
  IntType lcm(const IntType a, const IntType b) {
    return a / gcd(a, b) * b;
  }
  template <class ForwardIterator, class OutputIterator>
  void dissociate(ForwardIterator __begin, ForwardIterator __end,
                  OutputIterator __dest) {
    std::vector<typename std::iterator_traits<ForwardIterator>::_IntType> values(
        __begin, __end);
    std::sort(values.begin(), values.end());
    std::unique(values.begin(), values.end());
    while (__begin != __end) {
      *__dest =
          static_cast<typename std::iterator_traits<OutputIterator>::_IntType>(
              std::distance(
                  values.begin(),
                  std::lower_bound(values.begin(), values.end(), *__begin)));
      __dest++;
      __begin++;
    }
  }
  }  // namespace Temps
  using Temps::checkMax;
  using Temps::checkMin;
  namespace Temps {
  template <class Type>
  class Vector {
    static_assert(std::is_trivial<Type>::value,
                  "Temps::Vector can only be used for trival types");
  
   public:
    typedef Type& reference;
    typedef const Type& const_reference;
    class iterator : public std::iterator<std::random_access_iterator_tag, Type> {
     public:
      friend class Vector;
      using typename std::iterator<std::random_access_iterator_tag,
                                   Type>::difference_type;
      using
          typename std::iterator<std::random_access_iterator_tag, Type>::pointer;
      using typename std::iterator<std::random_access_iterator_tag,
                                   Type>::reference;
  
     private:
      pointer ptr;
      iterator(pointer ptr) : ptr(ptr) {}
  
     public:
      bool operator==(const iterator rhs) const { return ptr == rhs.ptr; }
      bool operator!=(const iterator rhs) const { return ptr != rhs.ptr; }
      bool operator<(const iterator rhs) const { return ptr < rhs.ptr; }
      bool operator<=(const iterator rhs) const { return ptr <= rhs.ptr; }
      bool operator>(const iterator rhs) const { return ptr > rhs.ptr; }
      bool operator>=(const iterator rhs) const { return ptr >= rhs.ptr; }
      iterator operator++() { return iterator(++ptr); }
      iterator operator++(int) { return iterator(ptr++); }
      iterator operator--() { return iterator(--ptr); }
      iterator operator--(int) { return iterator(ptr--); }
      iterator operator+(const difference_type dif) const {
        return iterator(ptr + dif);
      }
      iterator operator-(const difference_type dif) const {
        return iterator(ptr - dif);
      }
      iterator operator+=(const difference_type dif) {
        ptr += dif;
        return *this;
      }
      iterator operator-=(const difference_type dif) {
        ptr -= dif;
        return *this;
      }
      difference_type operator-(const iterator& rhs) const {
        return ptr - rhs.ptr;
      }
      operator pointer() { return ptr; }
    };
  
   protected:
    Type* a = nullptr;
    unsigned int _capacity = 0, _size = 0;
    void __grow_capacity(unsigned int least_size) {
      if (_capacity >= least_size) {
        return;
      }
      if (_capacity == 0) {
        _capacity = 1;
      }
      while (_capacity < least_size) {
        _capacity = _capacity * 2;
      }
      a = static_cast<Type*>(realloc(a, sizeof(Type) * _capacity));
    }
  
   public:
    Vector() = default;
    explicit Vector(unsigned int size, const Type& initial_value = Type())
        : _capacity(size), _size(size) {
      a = static_cast<Type*>(malloc(sizeof(Type) * _capacity));
      Type *__first = a, *const __last = a + _size;
      while (__first != __last) {
        *__first++ = initial_value;
      }
    }
    Vector(const Vector& rhs) : _capacity(rhs._size), _size(rhs._size) {
      a = static_cast<Type*>(malloc(sizeof(Type) * _capacity));
      Type *__first = a, *__r_p = rhs.a, *const __last = a + _size;
      while (__first != __last) {
        *__first++ = *__r_p++;
      }
    }
    Vector(Vector&& rhs) : _capacity(rhs._capacity), _size(rhs._size) {
      a = rhs.a;
      rhs.a = nullptr;
    }
    Vector(const std::initializer_list<Type>& list)
        : _capacity(
              static_cast<unsigned int>(std::distance(list.begin(), list.end()))),
          _size(_capacity) {
      a = static_cast<Type*>(malloc(sizeof(Type) * _capacity));
      Type* p = a;
      for (const auto& i : list) {
        *p++ = i;
      }
    }
    template <typename Container,
              typename = decltype(std::declval<Container>().begin(),
                                  std::declval<Container>().end())>
    Vector(const Container& list)
        : _capacity(
              static_cast<unsigned int>(std::distance(list.begin(), list.end()))),
          _size(_capacity) {
      a = static_cast<Type*>(malloc(sizeof(Type) * _capacity));
      Type* p = a;
      for (const auto& i : list) {
        *p++ = i;
      }
    }
    template <
        typename _InputIterator,
        typename = typename std::enable_if<std::is_convertible<
            typename std::iterator_traits<_InputIterator>::iterator_category,
            std::input_iterator_tag>::value>::type>
    explicit Vector(_InputIterator __first, _InputIterator __last) {
      _size = _capacity =
          static_cast<unsigned int>(std::distance(__first, __last));
      a = static_cast<Type*>(malloc(sizeof(Type) * _capacity));
      Type* p = a;
      while (__first != __last) {
        *p++ = *__first++;
      }
    }
    inline bool empty() const { return _size == 0; }
    inline unsigned int size() const { return _size; }
    inline unsigned int capacity() const { return _capacity; }
    void reserve(unsigned int __capacity) {
      _capacity = __capacity;
      if (_size > _capacity) {
        _size = _capacity;
      }
      a = static_cast<Type*>(realloc(a, sizeof(Type) * _capacity));
    }
    void resize(unsigned int __size) {
      if (_size > __size) {
        _size = __size;
        return;
      }
      __grow_capacity(__size);
      Type *__un = a + _size, *const __ed = a + __size;
      while (__un != __ed) {
        *__un++ = Type();
      }
      _size = __size;
    }
    inline void clear() { resize(0); }
    Vector& operator=(const Vector& rhs) {
      __grow_capacity(rhs._size);
      _size = rhs._size;
      Type* __first = a;
      const Type *const __last = a + _size, *__r_p = rhs.a;
      while (__first != __last) {
        *__first++ = *__r_p++;
      }
      return *this;
    }
    Vector& operator=(Vector&& rhs) {
      _size = rhs._size;
      _capacity = rhs._capacity;
      free(a);
      a = rhs.a;
      rhs.a = nullptr;
      return *this;
    }
    Type operator[](const unsigned int id) const { return a[id]; }
    Type& operator[](const unsigned int id) { return a[id]; }
    Type front() const { return a[0]; }
    Type& front() { return a[0]; }
    Type back() const { return a[_size - 1]; }
    Type& back() { return a[_size - 1]; }
    void pop_back() { _size--; }
    Vector& push_back(const Type& b) {
      __grow_capacity(_size + 1);
      a[_size] = b;
      _size++;
      return *this;
    }
    Vector& push_back(const Vector& rhs) {
      __grow_capacity(_size + rhs._size);
      for (const auto& i : rhs) {
        a[_size++] = i;
      }
      return *this;
    }
    Vector& push_back(const std::initializer_list<Type>& rhs) {
      __grow_capacity(_size + rhs.size());
      for (const auto& i : rhs) {
        a[_size++] = i;
      }
      return *this;
    }
    inline iterator begin() const { return iterator(a); }
    inline iterator end() const { return iterator(a + _size); }
    ~Vector() { free(a); }
  };
  }  // namespace Temps
  using Temps::Vector;
  void preInit();
  void init();
  void solve();
  int32_t main() {
    preInit();
    init();
    solve();
    return 0;
  }
  int area(int x1, int y1, int x2, int y2, int x3, int y3) {
    return (x1 * y2 - x1 * y3 - x2 * y1 + x2 * y3 + x3 * y1 - x3 * y2) & 3;
  }
  int boundary(int x1, int y1, int x2, int y2) {
    return Temps::gcd(std::abs(x1 - x2), std::abs(y1 - y2)) % 4;
  }
  constexpr int N = 6005;
  int x[N], y[N], cnt[N][4][4][4], n;
  void preInit() {}
  void init() {
    n = sc.n();
    for (int i = 0; i < n; i++) {
      x[i] = sc.n();
      y[i] = sc.n();
    }
    for (int i = 0; i < n; i++)
      for (int j = 0; j < n; j++)
        if (i != j)
          cnt[i][x[j] & 3][y[j] & 3][boundary(x[i], y[i], x[j], y[j])]++;
  }
  void solve() {
    long long eee_cnt = 0;
    long long ooe_cnt = 0;
    for (int i = 0; i < n; i++)
      for (int b1 = 0; b1 < 4; b1++)
        for (int b2 = b1; b2 < 4; b2 += 2)
          for (int s1 = 0; s1 < 16; s1++)
            for (int s2 = (b1 == b2 ? s1 : 0); s2 < 16; s2++) {
              int x1 = s1 / 4, y1 = s1 % 4;
              int x2 = s2 / 4, y2 = s2 % 4;
              if (x1 % 2 != x2 % 2 || y1 % 2 != y2 % 2) continue;
              int b3 = (x1 == x2 && y1 == y2) ? 0 : 2;
              int pcnt = (x1 == x2 && y1 == y2 && b1 == b2)
                             ? cnt[i][x1][y1][b1] * (cnt[i][x1][y1][b1] - 1) / 2
                             : cnt[i][x1][y1][b1] * cnt[i][x2][y2][b2];
              if (pcnt)
                ;
              if (area(x[i], y[i], x1, y1, x2, y2) == (b1 + b2 + b3) % 4)
                (b1 % 2 ? ooe_cnt : eee_cnt) += pcnt;
            };
    printf("%lld\n", ooe_cnt + eee_cnt / 3);
  }
  