#include <bits/stdc++.h>
using namespace std;
using int64 = long long;
enum Color { RED, BLACK };
template <typename T>
class RBNode {
 public:
  T key;
  RBNode *left, *right, *p;
  Color color;
  RBNode(const T& val)
      : key(val), left(nullptr), right(nullptr), p(nullptr), color(BLACK) {}
  friend ostream& operator<<(ostream& os, const RBNode& n) {
    os << n.key;
    return os;
  }
};
template <typename T>
class RBTree {
 public:
  RBNode<T>* root;

 public:
  RBTree() : root(nullptr) {}
  void print() const {
    if (!root) return;
    RBNode<T>* node = minimumNode();
    RBNode<T>* maxNode = maximumNode();
    cout << endl;
    while (node != maxNode) {
      cout << (*node) << " -> ";
      node = successor(node);
    }
    cout << (*node) << endl;
  }
  RBNode<T>* search(const T& val) const {
    if (!root) return nullptr;
    return iterativeSearch(root, val);
  }
  RBNode<T>* searchGreaterElement(const T& val) const {
    RBNode<T>* node = search(val);
    if (node->key < val) return successor(node);
    return node;
  }
  RBNode<T>* successor(RBNode<T>* x) const {
    if (!x) return nullptr;
    if (x->right) return minimumNode(x->right);
    RBNode<T>* y = x->p;
    while (y && x == y->right) {
      x = y;
      y = y->p;
    }
    return y;
  }
  RBNode<T>* predeccesor(RBNode<T>* x) const {
    if (!x) return nullptr;
    if (x->left) return maximumNode(x->left);
    RBNode<T>* y = x->p;
    while (y && x == y->left) {
      x = y;
      y = y->p;
    }
    return y;
  }
  void insert(RBNode<T>* z) {
    RBNode<T>* y = nullptr;
    RBNode<T>* x = root;
    while (x) {
      y = x;
      if (z->key < x->key)
        x = x->left;
      else
        x = x->right;
    }
    z->p = y;
    if (!y)
      root = z;
    else if (z->key < y->key)
      y->left = z;
    else
      y->right = z;
    z->left = nullptr;
    z->right = nullptr;
    setColor(z, RED);
    insertFixup(z);
  }
  RBNode<T>* erase(RBNode<T>* z) {
    RBNode<T>*x, *y;
    if (!(z->left) || !(z->right))
      y = z;
    else
      y = successor(z);
    if (y->left)
      x = y->left;
    else
      x = y->right;
    setParent(x, y->p);
    if (!(y->p))
      root = x;
    else if (y == y->p->left)
      y->p->left = x;
    else
      y->p->right = x;
    if (y != z) z->key = y->key;
    if (getColor(y) == BLACK && x) eraseFixup(x);
    return y;
  }
  RBNode<T>* minimumNode() const { return minimumNode(root); }
  RBNode<T>* maximumNode() const { return maximumNode(root); }

 private:
  inline Color getColor(RBNode<T>* x) const {
    if (!x) return BLACK;
    return x->color;
  }
  inline void setColor(RBNode<T>* x, Color color) const {
    if (x) x->color = color;
  }
  inline void setParent(RBNode<T>* x, RBNode<T>* p) const {
    if (x) x->p = p;
  }
  RBNode<T>* iterativeSearch(RBNode<T>* x, const T& val) const {
    while ((x->left || x->right) && val != x->key) {
      if (val < x->key) {
        if (!(x->left)) return x;
        x = x->left;
      } else {
        if (!(x->right)) return x;
        x = x->right;
      }
    }
    return x;
  }
  RBNode<T>* minimumNode(RBNode<T>* x) const {
    while (x->left) x = x->left;
    return x;
  }
  RBNode<T>* maximumNode(RBNode<T>* x) const {
    while (x->right) x = x->right;
    return x;
  }
  void leftRotate(RBNode<T>* x) {
    RBNode<T>* y = x->right;
    x->right = y->left;
    setParent(y->left, x);
    setParent(y, x->p);
    if (!(x->p))
      root = y;
    else if (x == x->p->left)
      x->p->left = y;
    else
      x->p->right = y;
    y->left = x;
    setParent(x, y);
  }
  void rightRotate(RBNode<T>* x) {
    RBNode<T>* y = x->left;
    x->left = y->right;
    setParent(y->right, x);
    setParent(y, x->p);
    if (!(x->p))
      root = y;
    else if (x == x->p->right)
      x->p->right = y;
    else
      x->p->left = y;
    y->right = x;
    setParent(x, y);
  }
  void insertFixup(RBNode<T>* z) {
    RBNode<T>* y;
    while (getColor(z->p) == RED) {
      if (z->p == z->p->p->left) {
        y = z->p->p->right;
        if (getColor(y) == RED) {
          setColor(z->p, BLACK);
          setColor(y, BLACK);
          setColor(z->p->p, RED);
          z = z->p->p;
        } else {
          if (z == z->p->right) {
            z = z->p;
            leftRotate(z);
          }
          setColor(z->p, BLACK);
          setColor(z->p->p, RED);
          rightRotate(z->p->p);
        }
      } else {
        y = z->p->p->left;
        if (getColor(y) == RED) {
          setColor(z->p, BLACK);
          setColor(y, BLACK);
          setColor(z->p->p, RED);
          z = z->p->p;
        } else {
          if (z == z->p->left) {
            z = z->p;
            rightRotate(z);
          }
          setColor(z->p, BLACK);
          setColor(z->p->p, RED);
          leftRotate(z->p->p);
        }
      }
    }
    setColor(root, BLACK);
  }
  void eraseFixup(RBNode<T>* x) {
    while (x != root && getColor(x) == BLACK) {
      if (x == x->p->left) {
        RBNode<T>* w = x->p->right;
        if (getColor(w) == RED) {
          setColor(w, BLACK);
          setColor(x->p, RED);
          leftRotate(x->p);
          w = x->p->right;
        }
        if (!w) break;
        if (getColor(w->left) == BLACK && getColor(w->right) == BLACK) {
          setColor(w, RED);
          x = x->p;
        } else {
          if (getColor(w->right) == BLACK) {
            setColor(w->left, BLACK);
            setColor(w, RED);
            rightRotate(w);
            w = x->p->right;
          }
          setColor(w, x->p->color);
          setColor(x->p, BLACK);
          setColor(w->right, BLACK);
          leftRotate(x->p);
          x = root;
        }
      } else {
        RBNode<T>* w = x->p->left;
        if (getColor(w) == RED) {
          setColor(w, BLACK);
          setColor(x->p, RED);
          rightRotate(x->p);
          w = x->p->left;
        }
        if (!w) break;
        if (getColor(w->right) == BLACK && getColor(w->left) == BLACK) {
          setColor(w, RED);
          x = x->p;
        } else {
          if (getColor(w->left) == BLACK) {
            setColor(w->right, BLACK);
            setColor(w, RED);
            leftRotate(w);
            w = x->p->left;
          }
          setColor(w, x->p->color);
          setColor(x->p, BLACK);
          setColor(w->left, BLACK);
          rightRotate(x->p);
          x = root;
        }
      }
    }
    setColor(x, BLACK);
  }
};
class Point;
int64 area2(const Point& p, const Point& q, const Point& s);
int toLeft(const Point& p, const Point& q, const Point& s);
class Point {
 public:
  int64 x, y;
  Point(int64 x_ = 0, int64 y_ = 0) : x(x_), y(y_) {}
  bool operator<(const Point& p) const {
    return x < p.x || (x == p.x && y < p.y);
  }
  bool operator>(const Point& p) const {
    return x > p.x || (x == p.x && y > p.y);
  }
  bool operator!=(const Point& p) const { return x != p.x || y != p.y; }
  friend ostream& operator<<(ostream& os, const Point& p) {
    os << "(" << p.x << ", " << p.y << ")";
    return os;
  }
};
int64 area2(const Point& p, const Point& q, const Point& s) {
  return p.x * q.y - p.y * q.x + q.x * s.y - q.y * s.x + s.x * p.y - s.y * p.x;
}
int toLeft(const Point& p, const Point& q, const Point& s) {
  int64 area = area2(p, q, s);
  if (area > 0)
    return 1;
  else if (area == 0)
    return 0;
  else
    return -1;
}
class ConvexHull {
 private:
  RBTree<Point> t;
  int idx;

 public:
  ConvexHull() : idx(0) {}
  void print() { t.print(); }
  void insert(const Point& p) {
    if (idx < 2) {
      idx++;
      t.insert(new RBNode<Point>(p));
      return;
    }
    if (inConvexHull(p)) return;
    RBNode<Point>* node = t.searchGreaterElement(Point(p.x, -1e9));
    if (node && node->key.x == p.x) t.erase(node);
    t.insert(new RBNode<Point>(p));
    RBNode<Point>* minimumNode = t.minimumNode();
    RBNode<Point>* maximumNode = t.maximumNode();
    RBNode<Point>*i, *j;
    node = t.search(p);
    i = t.predeccesor(node);
    if (i) {
      j = t.predeccesor(i);
      while (node->key != minimumNode->key && i->key != minimumNode->key) {
        if (toLeft(node->key, j->key, i->key) <= 0)
          t.erase(t.search(i->key));
        else
          break;
        i = t.search(j->key);
        j = t.predeccesor(i);
      }
    }
    node = t.search(p);
    i = t.successor(node);
    if (i) {
      j = t.successor(i);
      while (node->key != maximumNode->key && i->key != maximumNode->key) {
        if (toLeft(node->key, j->key, i->key) >= 0)
          t.erase(t.search(i->key));
        else
          break;
        i = t.search(j->key);
        j = t.successor(i);
      }
    }
  }
  bool inConvexHull(const Point& p) const {
    if (!t.root) return false;
    RBNode<Point>* minimumNode = t.minimumNode();
    RBNode<Point>* maximumNode = t.maximumNode();
    if (p.x < minimumNode->key.x || p.x > maximumNode->key.x) return false;
    RBNode<Point>* node = t.searchGreaterElement(Point(p.x, -1e9));
    if (node && node->key.x == p.x) return p.y >= node->key.y;
    RBNode<Point>* r = t.searchGreaterElement(p);
    RBNode<Point>* l = t.predeccesor(r);
    return toLeft(p, l->key, r->key) >= 0;
  }
};
int main() {
  int n;
  scanf("%d", &n);
  ConvexHull chUp, chDown;
  for (int i = 0; i < n; ++i) {
    int op;
    int x, y;
    scanf("%d%d%d", &op, &x, &y);
    Point pUp(x, -y), pDown(x, y);
    if (op == 1) {
      chUp.insert(pUp);
      chDown.insert(pDown);
    } else {
      if (chUp.inConvexHull(pUp) && chDown.inConvexHull(pDown))
        printf("YES\n");
      else
        printf("NO\n");
    }
  }
  return 0;
}

### Prompt

Your challenge is to write a CPP solution to the following problem:

Once a walrus professor Plato asked his programming students to perform the following practical task. 

The students had to implement such a data structure that would support a convex hull on some set of points S. The input to the program had q queries of two types: 

1. Add a point with coordinates (x, y) into the set S. Note that in this case the convex hull of S could have changed, and could have remained the same. 

2. Say whether a point with coordinates (x, y) belongs to an area limited by the convex hull, including the border. 

All the students coped with the task. What about you?

Input

The first line contains an integer q (4 ≤ q ≤ 105). 

Then follow q lines in the following way: "t x y", where t is the query type (1 or 2), and (x, y) are the coordinates of the point ( - 106 ≤ x, y ≤ 106, x and y are integers). 

There is at least one query of type 2.

It is guaranteed that the three queries of the first type follow first and the points given in the queries form a non-degenerative triangle. Also all the points added in S are distinct.

Output

For each query of the second type print one string containing "YES", if the point lies inside the convex hull or on its border. Otherwise, print "NO".

Examples

Input

8
1 0 0
1 2 0
1 2 2
2 1 0
1 0 2
2 1 1
2 2 1
2 20 -1


Output

YES
YES
YES
NO

### Response

```cpp
#include <bits/stdc++.h>
using namespace std;
using int64 = long long;
enum Color { RED, BLACK };
template <typename T>
class RBNode {
 public:
  T key;
  RBNode *left, *right, *p;
  Color color;
  RBNode(const T& val)
      : key(val), left(nullptr), right(nullptr), p(nullptr), color(BLACK) {}
  friend ostream& operator<<(ostream& os, const RBNode& n) {
    os << n.key;
    return os;
  }
};
template <typename T>
class RBTree {
 public:
  RBNode<T>* root;

 public:
  RBTree() : root(nullptr) {}
  void print() const {
    if (!root) return;
    RBNode<T>* node = minimumNode();
    RBNode<T>* maxNode = maximumNode();
    cout << endl;
    while (node != maxNode) {
      cout << (*node) << " -> ";
      node = successor(node);
    }
    cout << (*node) << endl;
  }
  RBNode<T>* search(const T& val) const {
    if (!root) return nullptr;
    return iterativeSearch(root, val);
  }
  RBNode<T>* searchGreaterElement(const T& val) const {
    RBNode<T>* node = search(val);
    if (node->key < val) return successor(node);
    return node;
  }
  RBNode<T>* successor(RBNode<T>* x) const {
    if (!x) return nullptr;
    if (x->right) return minimumNode(x->right);
    RBNode<T>* y = x->p;
    while (y && x == y->right) {
      x = y;
      y = y->p;
    }
    return y;
  }
  RBNode<T>* predeccesor(RBNode<T>* x) const {
    if (!x) return nullptr;
    if (x->left) return maximumNode(x->left);
    RBNode<T>* y = x->p;
    while (y && x == y->left) {
      x = y;
      y = y->p;
    }
    return y;
  }
  void insert(RBNode<T>* z) {
    RBNode<T>* y = nullptr;
    RBNode<T>* x = root;
    while (x) {
      y = x;
      if (z->key < x->key)
        x = x->left;
      else
        x = x->right;
    }
    z->p = y;
    if (!y)
      root = z;
    else if (z->key < y->key)
      y->left = z;
    else
      y->right = z;
    z->left = nullptr;
    z->right = nullptr;
    setColor(z, RED);
    insertFixup(z);
  }
  RBNode<T>* erase(RBNode<T>* z) {
    RBNode<T>*x, *y;
    if (!(z->left) || !(z->right))
      y = z;
    else
      y = successor(z);
    if (y->left)
      x = y->left;
    else
      x = y->right;
    setParent(x, y->p);
    if (!(y->p))
      root = x;
    else if (y == y->p->left)
      y->p->left = x;
    else
      y->p->right = x;
    if (y != z) z->key = y->key;
    if (getColor(y) == BLACK && x) eraseFixup(x);
    return y;
  }
  RBNode<T>* minimumNode() const { return minimumNode(root); }
  RBNode<T>* maximumNode() const { return maximumNode(root); }

 private:
  inline Color getColor(RBNode<T>* x) const {
    if (!x) return BLACK;
    return x->color;
  }
  inline void setColor(RBNode<T>* x, Color color) const {
    if (x) x->color = color;
  }
  inline void setParent(RBNode<T>* x, RBNode<T>* p) const {
    if (x) x->p = p;
  }
  RBNode<T>* iterativeSearch(RBNode<T>* x, const T& val) const {
    while ((x->left || x->right) && val != x->key) {
      if (val < x->key) {
        if (!(x->left)) return x;
        x = x->left;
      } else {
        if (!(x->right)) return x;
        x = x->right;
      }
    }
    return x;
  }
  RBNode<T>* minimumNode(RBNode<T>* x) const {
    while (x->left) x = x->left;
    return x;
  }
  RBNode<T>* maximumNode(RBNode<T>* x) const {
    while (x->right) x = x->right;
    return x;
  }
  void leftRotate(RBNode<T>* x) {
    RBNode<T>* y = x->right;
    x->right = y->left;
    setParent(y->left, x);
    setParent(y, x->p);
    if (!(x->p))
      root = y;
    else if (x == x->p->left)
      x->p->left = y;
    else
      x->p->right = y;
    y->left = x;
    setParent(x, y);
  }
  void rightRotate(RBNode<T>* x) {
    RBNode<T>* y = x->left;
    x->left = y->right;
    setParent(y->right, x);
    setParent(y, x->p);
    if (!(x->p))
      root = y;
    else if (x == x->p->right)
      x->p->right = y;
    else
      x->p->left = y;
    y->right = x;
    setParent(x, y);
  }
  void insertFixup(RBNode<T>* z) {
    RBNode<T>* y;
    while (getColor(z->p) == RED) {
      if (z->p == z->p->p->left) {
        y = z->p->p->right;
        if (getColor(y) == RED) {
          setColor(z->p, BLACK);
          setColor(y, BLACK);
          setColor(z->p->p, RED);
          z = z->p->p;
        } else {
          if (z == z->p->right) {
            z = z->p;
            leftRotate(z);
          }
          setColor(z->p, BLACK);
          setColor(z->p->p, RED);
          rightRotate(z->p->p);
        }
      } else {
        y = z->p->p->left;
        if (getColor(y) == RED) {
          setColor(z->p, BLACK);
          setColor(y, BLACK);
          setColor(z->p->p, RED);
          z = z->p->p;
        } else {
          if (z == z->p->left) {
            z = z->p;
            rightRotate(z);
          }
          setColor(z->p, BLACK);
          setColor(z->p->p, RED);
          leftRotate(z->p->p);
        }
      }
    }
    setColor(root, BLACK);
  }
  void eraseFixup(RBNode<T>* x) {
    while (x != root && getColor(x) == BLACK) {
      if (x == x->p->left) {
        RBNode<T>* w = x->p->right;
        if (getColor(w) == RED) {
          setColor(w, BLACK);
          setColor(x->p, RED);
          leftRotate(x->p);
          w = x->p->right;
        }
        if (!w) break;
        if (getColor(w->left) == BLACK && getColor(w->right) == BLACK) {
          setColor(w, RED);
          x = x->p;
        } else {
          if (getColor(w->right) == BLACK) {
            setColor(w->left, BLACK);
            setColor(w, RED);
            rightRotate(w);
            w = x->p->right;
          }
          setColor(w, x->p->color);
          setColor(x->p, BLACK);
          setColor(w->right, BLACK);
          leftRotate(x->p);
          x = root;
        }
      } else {
        RBNode<T>* w = x->p->left;
        if (getColor(w) == RED) {
          setColor(w, BLACK);
          setColor(x->p, RED);
          rightRotate(x->p);
          w = x->p->left;
        }
        if (!w) break;
        if (getColor(w->right) == BLACK && getColor(w->left) == BLACK) {
          setColor(w, RED);
          x = x->p;
        } else {
          if (getColor(w->left) == BLACK) {
            setColor(w->right, BLACK);
            setColor(w, RED);
            leftRotate(w);
            w = x->p->left;
          }
          setColor(w, x->p->color);
          setColor(x->p, BLACK);
          setColor(w->left, BLACK);
          rightRotate(x->p);
          x = root;
        }
      }
    }
    setColor(x, BLACK);
  }
};
class Point;
int64 area2(const Point& p, const Point& q, const Point& s);
int toLeft(const Point& p, const Point& q, const Point& s);
class Point {
 public:
  int64 x, y;
  Point(int64 x_ = 0, int64 y_ = 0) : x(x_), y(y_) {}
  bool operator<(const Point& p) const {
    return x < p.x || (x == p.x && y < p.y);
  }
  bool operator>(const Point& p) const {
    return x > p.x || (x == p.x && y > p.y);
  }
  bool operator!=(const Point& p) const { return x != p.x || y != p.y; }
  friend ostream& operator<<(ostream& os, const Point& p) {
    os << "(" << p.x << ", " << p.y << ")";
    return os;
  }
};
int64 area2(const Point& p, const Point& q, const Point& s) {
  return p.x * q.y - p.y * q.x + q.x * s.y - q.y * s.x + s.x * p.y - s.y * p.x;
}
int toLeft(const Point& p, const Point& q, const Point& s) {
  int64 area = area2(p, q, s);
  if (area > 0)
    return 1;
  else if (area == 0)
    return 0;
  else
    return -1;
}
class ConvexHull {
 private:
  RBTree<Point> t;
  int idx;

 public:
  ConvexHull() : idx(0) {}
  void print() { t.print(); }
  void insert(const Point& p) {
    if (idx < 2) {
      idx++;
      t.insert(new RBNode<Point>(p));
      return;
    }
    if (inConvexHull(p)) return;
    RBNode<Point>* node = t.searchGreaterElement(Point(p.x, -1e9));
    if (node && node->key.x == p.x) t.erase(node);
    t.insert(new RBNode<Point>(p));
    RBNode<Point>* minimumNode = t.minimumNode();
    RBNode<Point>* maximumNode = t.maximumNode();
    RBNode<Point>*i, *j;
    node = t.search(p);
    i = t.predeccesor(node);
    if (i) {
      j = t.predeccesor(i);
      while (node->key != minimumNode->key && i->key != minimumNode->key) {
        if (toLeft(node->key, j->key, i->key) <= 0)
          t.erase(t.search(i->key));
        else
          break;
        i = t.search(j->key);
        j = t.predeccesor(i);
      }
    }
    node = t.search(p);
    i = t.successor(node);
    if (i) {
      j = t.successor(i);
      while (node->key != maximumNode->key && i->key != maximumNode->key) {
        if (toLeft(node->key, j->key, i->key) >= 0)
          t.erase(t.search(i->key));
        else
          break;
        i = t.search(j->key);
        j = t.successor(i);
      }
    }
  }
  bool inConvexHull(const Point& p) const {
    if (!t.root) return false;
    RBNode<Point>* minimumNode = t.minimumNode();
    RBNode<Point>* maximumNode = t.maximumNode();
    if (p.x < minimumNode->key.x || p.x > maximumNode->key.x) return false;
    RBNode<Point>* node = t.searchGreaterElement(Point(p.x, -1e9));
    if (node && node->key.x == p.x) return p.y >= node->key.y;
    RBNode<Point>* r = t.searchGreaterElement(p);
    RBNode<Point>* l = t.predeccesor(r);
    return toLeft(p, l->key, r->key) >= 0;
  }
};
int main() {
  int n;
  scanf("%d", &n);
  ConvexHull chUp, chDown;
  for (int i = 0; i < n; ++i) {
    int op;
    int x, y;
    scanf("%d%d%d", &op, &x, &y);
    Point pUp(x, -y), pDown(x, y);
    if (op == 1) {
      chUp.insert(pUp);
      chDown.insert(pDown);
    } else {
      if (chUp.inConvexHull(pUp) && chDown.inConvexHull(pDown))
        printf("YES\n");
      else
        printf("NO\n");
    }
  }
  return 0;
}

```