#include <bits/stdc++.h>
using namespace std;
const int MAXN = 1e5 + 10;
const int INF32 = 0x3f3f3f3f;
const long long INF64 = 0x3f3f3f3f3f3f3f3f;
const long long mod = 1e9 + 7;
const double PI = acos(-1.0);
const double EPS = 1.0e-8;
struct Point {
  double x, y, t, d;
  Point(double _x = 0, double _y = 0, double _t = 0, double _d = 0) {
    x = _x;
    y = _y;
    t = _t;
    d = _d;
  }
  friend Point operator+(const Point &a, const Point &b) {
    return Point(a.x + b.x, a.y + b.y);
  }
  friend Point operator-(const Point &a, const Point &b) {
    return Point(a.x - b.x, a.y - b.y);
  }
  friend double operator^(Point a, Point b) { return a.x * b.y - a.y * b.x; }
  friend int operator==(const Point &a, const Point &b) {
    if (fabs(a.x - b.x) < EPS && fabs(a.y - b.y) < EPS) return 1;
    return 0;
  }
  friend double operator*(const Point &a, const Point &b) {
    return a.x * b.x + a.y * b.y;
  }
};
struct V {
  Point start, end;
  double ang;
  V(Point _start = Point(0, 0), Point _end = Point(0, 0), double _ang = 0.0) {
    start = _start;
    end = _end;
    ang = _ang;
  }
  friend V operator+(const V &a, const V &b) {
    return V(a.start + b.start, a.end + b.end);
  }
  friend V operator-(const V &a, const V &b) {
    return V(a.start - b.start, a.end - b.end);
  }
};
Point Basic, points[MAXN];
set<Point> S;
int N;
int Sgn(double key) { return fabs(key) < EPS ? 0 : (key < 0 ? -1 : 1); }
int Cmp(double key1, double key2) { return Sgn(key1 - key2); }
double GetLen(Point key) { return sqrt(key * key); }
bool operator<(Point key1, Point key2) {
  key1 = key1 - Basic;
  key2 = key2 - Basic;
  double Ang1 = atan2(key1.y, key1.x), Ang2 = atan2(key2.y, key2.x);
  double Len1 = GetLen(key1), Len2 = GetLen(key2);
  if (Cmp(Ang1, Ang2) != 0) return Cmp(Ang1, Ang2) < 0;
  return Cmp(Len1, Len2) < 0;
}
set<Point>::iterator Prev(set<Point>::iterator key) {
  if (key == S.begin()) key = S.end();
  return --key;
}
set<Point>::iterator Next(set<Point>::iterator key) {
  ++key;
  return key == S.end() ? S.begin() : key;
}
int Query(Point key) {
  set<Point>::iterator it = S.lower_bound(key);
  if (it == S.end()) it = S.begin();
  return Sgn((key - *(Prev(it))) ^ (*(it) - *(Prev(it)))) <= 0;
}
void Insert(Point key) {
  if (Query(key)) return;
  S.insert(key);
  set<Point>::iterator Cur = Next(S.find(key));
  while (S.size() > 3 &&
         Sgn((key - *(Next(Cur))) ^ (*(Cur) - *(Next(Cur)))) <= 0) {
    S.erase(Cur);
    Cur = Next(S.find(key));
  }
  Cur = Prev(S.find(key));
  while (S.size() > 3 && Sgn((key - *(Cur)) ^ (*(Cur) - *(Prev(Cur)))) >= 0) {
    S.erase(Cur);
    Cur = Prev(S.find(key));
  }
}
int main() {
  scanf("%d", &N);
  Basic = Point(0, 0);
  for (int i = 1, T; i <= 3; ++i) {
    scanf("%d%lf%lf", &T, &points[i].x, &points[i].y);
    Basic = Basic + points[i];
  }
  Basic.x /= 3.0;
  Basic.y /= 3.0;
  for (int i = 1; i <= 3; ++i) {
    S.insert(points[i]);
  }
  for (int i = 4, T; i <= N; ++i) {
    scanf("%d%lf%lf", &T, &points[i].x, &points[i].y);
    if (T == 1)
      Insert(points[i]);
    else {
      if (Query(points[i]))
        printf("YES\n");
      else
        printf("NO\n");
    }
  }
}

### Prompt

Construct a CPP code solution to the problem outlined:

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;
const int MAXN = 1e5 + 10;
const int INF32 = 0x3f3f3f3f;
const long long INF64 = 0x3f3f3f3f3f3f3f3f;
const long long mod = 1e9 + 7;
const double PI = acos(-1.0);
const double EPS = 1.0e-8;
struct Point {
  double x, y, t, d;
  Point(double _x = 0, double _y = 0, double _t = 0, double _d = 0) {
    x = _x;
    y = _y;
    t = _t;
    d = _d;
  }
  friend Point operator+(const Point &a, const Point &b) {
    return Point(a.x + b.x, a.y + b.y);
  }
  friend Point operator-(const Point &a, const Point &b) {
    return Point(a.x - b.x, a.y - b.y);
  }
  friend double operator^(Point a, Point b) { return a.x * b.y - a.y * b.x; }
  friend int operator==(const Point &a, const Point &b) {
    if (fabs(a.x - b.x) < EPS && fabs(a.y - b.y) < EPS) return 1;
    return 0;
  }
  friend double operator*(const Point &a, const Point &b) {
    return a.x * b.x + a.y * b.y;
  }
};
struct V {
  Point start, end;
  double ang;
  V(Point _start = Point(0, 0), Point _end = Point(0, 0), double _ang = 0.0) {
    start = _start;
    end = _end;
    ang = _ang;
  }
  friend V operator+(const V &a, const V &b) {
    return V(a.start + b.start, a.end + b.end);
  }
  friend V operator-(const V &a, const V &b) {
    return V(a.start - b.start, a.end - b.end);
  }
};
Point Basic, points[MAXN];
set<Point> S;
int N;
int Sgn(double key) { return fabs(key) < EPS ? 0 : (key < 0 ? -1 : 1); }
int Cmp(double key1, double key2) { return Sgn(key1 - key2); }
double GetLen(Point key) { return sqrt(key * key); }
bool operator<(Point key1, Point key2) {
  key1 = key1 - Basic;
  key2 = key2 - Basic;
  double Ang1 = atan2(key1.y, key1.x), Ang2 = atan2(key2.y, key2.x);
  double Len1 = GetLen(key1), Len2 = GetLen(key2);
  if (Cmp(Ang1, Ang2) != 0) return Cmp(Ang1, Ang2) < 0;
  return Cmp(Len1, Len2) < 0;
}
set<Point>::iterator Prev(set<Point>::iterator key) {
  if (key == S.begin()) key = S.end();
  return --key;
}
set<Point>::iterator Next(set<Point>::iterator key) {
  ++key;
  return key == S.end() ? S.begin() : key;
}
int Query(Point key) {
  set<Point>::iterator it = S.lower_bound(key);
  if (it == S.end()) it = S.begin();
  return Sgn((key - *(Prev(it))) ^ (*(it) - *(Prev(it)))) <= 0;
}
void Insert(Point key) {
  if (Query(key)) return;
  S.insert(key);
  set<Point>::iterator Cur = Next(S.find(key));
  while (S.size() > 3 &&
         Sgn((key - *(Next(Cur))) ^ (*(Cur) - *(Next(Cur)))) <= 0) {
    S.erase(Cur);
    Cur = Next(S.find(key));
  }
  Cur = Prev(S.find(key));
  while (S.size() > 3 && Sgn((key - *(Cur)) ^ (*(Cur) - *(Prev(Cur)))) >= 0) {
    S.erase(Cur);
    Cur = Prev(S.find(key));
  }
}
int main() {
  scanf("%d", &N);
  Basic = Point(0, 0);
  for (int i = 1, T; i <= 3; ++i) {
    scanf("%d%lf%lf", &T, &points[i].x, &points[i].y);
    Basic = Basic + points[i];
  }
  Basic.x /= 3.0;
  Basic.y /= 3.0;
  for (int i = 1; i <= 3; ++i) {
    S.insert(points[i]);
  }
  for (int i = 4, T; i <= N; ++i) {
    scanf("%d%lf%lf", &T, &points[i].x, &points[i].y);
    if (T == 1)
      Insert(points[i]);
    else {
      if (Query(points[i]))
        printf("YES\n");
      else
        printf("NO\n");
    }
  }
}

```