#include <bits/stdc++.h>

const int N=210,Mod=998244353;

struct Point{
	int x,y;
	Point(int x=0,int y=0):x(x),y(y) {}
	Point operator - ( const Point &A ) const { return Point(x-A.x,y-A.y); }
	Point operator / ( const int &p ) const { return Point(x/p,y/p); }
	bool operator == ( const Point &A ) const { return x==A.x && y==A.y; }
	bool operator < ( const Point &A ) const { return x<A.x || ( x==A.x && y<A.y); }
}p[N],q[N];
typedef Point Vector;

int n,qp2[N];

void Init() {
	scanf("%d",&n);
	for (int i=1;i<=n;++i) scanf("%d%d",&p[i].x,&p[i].y);
}

int gcd(int a,int b) { return (!b)?a:gcd(b,a%b); }

Vector C(const Vector &a) { return a/gcd(a.x,a.y); }

void Solve() {
	int ans=1; 
	qp2[0]=1; for (int i=1;i<=n;++i) qp2[i]=(qp2[i-1]<<1)%Mod,ans=(ans<<1)%Mod;
	ans--; ans-=n;
	std::sort(p+1,p+n+1);
	for (int i=1;i<=n;++i) {
		int t=0;
		for (int j=i+1;j<=n;++j) 
			q[++t]=C(p[j]-p[i]);
		std::sort(q+1,q+t+1);
		for (int h=1;h<=t;) {
			int p=h; while (q[p]==q[h] && h<=t) ++h;
			ans=(0ll+ans-(qp2[h-p]-1)+Mod)%Mod;
		}
	}
	printf("%d\n",ans);
}

int main() {
	Init();
	Solve();
	return 0;
}
### Prompt

Develop a solution in CPP to the problem described below:

You are given N points (x_i,y_i) located on a two-dimensional plane. Consider a subset S of the N points that forms a convex polygon. Here, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.

cddb0c267926c2add885ca153c47ad8a.png

For example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.

For a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.

Compute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.

However, since the sum can be extremely large, print the sum modulo 998244353.

Constraints

* 1≤N≤200
* 0≤x_i,y_i<10^4 (1≤i≤N)
* If i≠j, x_i≠x_j or y_i≠y_j.
* x_i and y_i are integers.

Input

The input is given from Standard Input in the following format:


N
x_1 y_1
x_2 y_2
:
x_N y_N


Output

Print the sum of all the scores modulo 998244353.

Examples

Input

4
0 0
0 1
1 0
1 1


Output

5


Input

5
0 0
0 1
0 2
0 3
1 1


Output

11


Input

1
3141 2718


Output

0

### Response

```cpp
#include <bits/stdc++.h>

const int N=210,Mod=998244353;

struct Point{
	int x,y;
	Point(int x=0,int y=0):x(x),y(y) {}
	Point operator - ( const Point &A ) const { return Point(x-A.x,y-A.y); }
	Point operator / ( const int &p ) const { return Point(x/p,y/p); }
	bool operator == ( const Point &A ) const { return x==A.x && y==A.y; }
	bool operator < ( const Point &A ) const { return x<A.x || ( x==A.x && y<A.y); }
}p[N],q[N];
typedef Point Vector;

int n,qp2[N];

void Init() {
	scanf("%d",&n);
	for (int i=1;i<=n;++i) scanf("%d%d",&p[i].x,&p[i].y);
}

int gcd(int a,int b) { return (!b)?a:gcd(b,a%b); }

Vector C(const Vector &a) { return a/gcd(a.x,a.y); }

void Solve() {
	int ans=1; 
	qp2[0]=1; for (int i=1;i<=n;++i) qp2[i]=(qp2[i-1]<<1)%Mod,ans=(ans<<1)%Mod;
	ans--; ans-=n;
	std::sort(p+1,p+n+1);
	for (int i=1;i<=n;++i) {
		int t=0;
		for (int j=i+1;j<=n;++j) 
			q[++t]=C(p[j]-p[i]);
		std::sort(q+1,q+t+1);
		for (int h=1;h<=t;) {
			int p=h; while (q[p]==q[h] && h<=t) ++h;
			ans=(0ll+ans-(qp2[h-p]-1)+Mod)%Mod;
		}
	}
	printf("%d\n",ans);
}

int main() {
	Init();
	Solve();
	return 0;
}
```