#include <bits/stdc++.h>
#pragma GCC optimize(4)
using namespace std;
char _buf[100000], *_p1 = _buf, *_p2 = _buf;
inline int gi() {
  int x = 0, f = 1;
  char ch = getchar();
  while (ch < '0' || ch > '9') {
    if (ch == '-') f = -1;
    ch = getchar();
  }
  while (ch >= '0' && ch <= '9') {
    x = (x << 3) + (x << 1) + (ch ^ 48);
    ch = getchar();
  }
  return (f == 1) ? x : -x;
}
inline int max(int a, int b) { return a > b ? a : b; }
inline int min(int a, int b) { return a < b ? a : b; }
const double eps = 1e-8;
const int maxn = 1e5 + 5;
inline int sign(double x) { return x < eps ? (x > -eps ? 0 : -1) : 1; }
struct Point {
  double x, y;
  Point(double xx = 0, double yy = 0) { x = xx, y = yy; }
  inline bool operator<(const Point b) const {
    return sign(x - b.x) == 0 ? y < b.y : x < b.x;
  }
  inline Point operator+(Point b) { return Point(x + b.x, y + b.y); }
  inline Point operator-(Point b) { return Point(x - b.x, y - b.y); }
  inline Point operator*(double b) { return Point(x * b, y * b); }
  inline double operator*(Point b) { return x * b.y - y * b.x; }
};
struct tubao {
  set<Point> up, down;
  inline bool check_up(Point x) {
    Point temp = x;
    temp.y = -1e9;
    set<Point>::iterator it = up.lower_bound(temp);
    if (it == up.end()) return false;
    set<Point>::iterator temp1 = it;
    temp1++;
    if (sign((*it).x - x.x) == 0) return x.y <= (*it).y;
    if (it == up.begin()) return false;
    set<Point>::iterator jt = it;
    --jt;
    Point i = *it, j = *jt;
    return sign((i - j) * (x - j)) <= 0;
  }
  inline bool check_down(Point x) {
    Point temp = x;
    temp.y = -1e9;
    set<Point>::iterator it = down.lower_bound(temp);
    if (it == down.end()) return false;
    if (sign((*it).x - x.x) == 0) return x.y >= (*it).y;
    if (it == down.begin()) return false;
    set<Point>::iterator jt = it;
    --jt;
    Point i = *it, j = *jt;
    return sign((i - j) * (x - j)) >= 0;
  }
  inline void insert_up(Point x) {
    if (check_up(x)) return;
    set<Point>::iterator temp = up.lower_bound(Point(x.x, -1e9));
    if (temp != up.end() && sign((*temp).x - x.x) == 0) up.erase(temp);
    up.insert(x);
    set<Point>::iterator it = up.find(x);
    set<Point>::iterator jt = it;
    if (it != up.begin()) {
      --jt;
      while (remove_up(jt++)) jt--;
    }
    if (++jt != up.end()) {
      while (remove_up(jt--)) jt++;
    }
  }
  inline void insert_down(Point x) {
    if (check_down(x)) return;
    set<Point>::iterator temp = down.lower_bound(Point(x.x, -1e9));
    if (temp != down.end() && sign((*temp).x - x.x) == 0) down.erase(temp);
    down.insert(x);
    set<Point>::iterator it = down.find(x);
    set<Point>::iterator jt = it;
    if (it != down.begin()) {
      --jt;
      while (remove_down(jt++)) jt--;
    }
    if (++jt != down.end()) {
      while (remove_down(jt--)) jt++;
    }
  }
  inline bool remove_up(set<Point>::iterator it) {
    if (it == up.begin()) return false;
    set<Point>::iterator jt = it, kt = it;
    --jt, ++kt;
    if (kt == up.end()) return false;
    Point i = *it, j = *jt, k = *kt;
    if ((i - j) * (k - j) >= 0) return up.erase(it), true;
    return false;
  }
  inline bool remove_down(set<Point>::iterator it) {
    if (it == down.begin()) return false;
    set<Point>::iterator jt = it, kt = it;
    jt--;
    kt++;
    if (kt == down.end()) return false;
    Point i = *it, j = *jt, k = *kt;
    if ((i - j) * (k - j) <= 0) return down.erase(it), true;
    return false;
  }
  inline void insert(Point x) {
    insert_up(x);
    insert_down(x);
  }
  inline bool query(Point x) { return check_up(x) && check_down(x); }
} tree;
inline void init() {}
inline void input() {}
inline void solve() {
  int Q = gi();
  while (Q--) {
    int t = gi(), x = gi(), y = gi();
    if (t == 1)
      tree.insert(Point(x, y));
    else
      printf(tree.query(Point(x, y)) ? "YES\n" : "NO\n");
  }
}
int main() {
  int T = 1;
  while (T--) {
    init();
    input();
    solve();
  }
  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>
#pragma GCC optimize(4)
using namespace std;
char _buf[100000], *_p1 = _buf, *_p2 = _buf;
inline int gi() {
  int x = 0, f = 1;
  char ch = getchar();
  while (ch < '0' || ch > '9') {
    if (ch == '-') f = -1;
    ch = getchar();
  }
  while (ch >= '0' && ch <= '9') {
    x = (x << 3) + (x << 1) + (ch ^ 48);
    ch = getchar();
  }
  return (f == 1) ? x : -x;
}
inline int max(int a, int b) { return a > b ? a : b; }
inline int min(int a, int b) { return a < b ? a : b; }
const double eps = 1e-8;
const int maxn = 1e5 + 5;
inline int sign(double x) { return x < eps ? (x > -eps ? 0 : -1) : 1; }
struct Point {
  double x, y;
  Point(double xx = 0, double yy = 0) { x = xx, y = yy; }
  inline bool operator<(const Point b) const {
    return sign(x - b.x) == 0 ? y < b.y : x < b.x;
  }
  inline Point operator+(Point b) { return Point(x + b.x, y + b.y); }
  inline Point operator-(Point b) { return Point(x - b.x, y - b.y); }
  inline Point operator*(double b) { return Point(x * b, y * b); }
  inline double operator*(Point b) { return x * b.y - y * b.x; }
};
struct tubao {
  set<Point> up, down;
  inline bool check_up(Point x) {
    Point temp = x;
    temp.y = -1e9;
    set<Point>::iterator it = up.lower_bound(temp);
    if (it == up.end()) return false;
    set<Point>::iterator temp1 = it;
    temp1++;
    if (sign((*it).x - x.x) == 0) return x.y <= (*it).y;
    if (it == up.begin()) return false;
    set<Point>::iterator jt = it;
    --jt;
    Point i = *it, j = *jt;
    return sign((i - j) * (x - j)) <= 0;
  }
  inline bool check_down(Point x) {
    Point temp = x;
    temp.y = -1e9;
    set<Point>::iterator it = down.lower_bound(temp);
    if (it == down.end()) return false;
    if (sign((*it).x - x.x) == 0) return x.y >= (*it).y;
    if (it == down.begin()) return false;
    set<Point>::iterator jt = it;
    --jt;
    Point i = *it, j = *jt;
    return sign((i - j) * (x - j)) >= 0;
  }
  inline void insert_up(Point x) {
    if (check_up(x)) return;
    set<Point>::iterator temp = up.lower_bound(Point(x.x, -1e9));
    if (temp != up.end() && sign((*temp).x - x.x) == 0) up.erase(temp);
    up.insert(x);
    set<Point>::iterator it = up.find(x);
    set<Point>::iterator jt = it;
    if (it != up.begin()) {
      --jt;
      while (remove_up(jt++)) jt--;
    }
    if (++jt != up.end()) {
      while (remove_up(jt--)) jt++;
    }
  }
  inline void insert_down(Point x) {
    if (check_down(x)) return;
    set<Point>::iterator temp = down.lower_bound(Point(x.x, -1e9));
    if (temp != down.end() && sign((*temp).x - x.x) == 0) down.erase(temp);
    down.insert(x);
    set<Point>::iterator it = down.find(x);
    set<Point>::iterator jt = it;
    if (it != down.begin()) {
      --jt;
      while (remove_down(jt++)) jt--;
    }
    if (++jt != down.end()) {
      while (remove_down(jt--)) jt++;
    }
  }
  inline bool remove_up(set<Point>::iterator it) {
    if (it == up.begin()) return false;
    set<Point>::iterator jt = it, kt = it;
    --jt, ++kt;
    if (kt == up.end()) return false;
    Point i = *it, j = *jt, k = *kt;
    if ((i - j) * (k - j) >= 0) return up.erase(it), true;
    return false;
  }
  inline bool remove_down(set<Point>::iterator it) {
    if (it == down.begin()) return false;
    set<Point>::iterator jt = it, kt = it;
    jt--;
    kt++;
    if (kt == down.end()) return false;
    Point i = *it, j = *jt, k = *kt;
    if ((i - j) * (k - j) <= 0) return down.erase(it), true;
    return false;
  }
  inline void insert(Point x) {
    insert_up(x);
    insert_down(x);
  }
  inline bool query(Point x) { return check_up(x) && check_down(x); }
} tree;
inline void init() {}
inline void input() {}
inline void solve() {
  int Q = gi();
  while (Q--) {
    int t = gi(), x = gi(), y = gi();
    if (t == 1)
      tree.insert(Point(x, y));
    else
      printf(tree.query(Point(x, y)) ? "YES\n" : "NO\n");
  }
}
int main() {
  int T = 1;
  while (T--) {
    init();
    input();
    solve();
  }
  return 0;
}

```