#include<bits/stdc++.h>
using namespace std;
const int P=998244353;
int n;
struct pos{
	int x,y;
}ps[1007];
int ep=0,f[1007],sz[1007];
int gf(int x){while(x!=f[x])x=f[x]=x[f][f];return x;}
struct edge{
	int a,b;
	int v;
	bool operator<(const edge&e)const{return v<e.v;}
	void cal(){
		int x=gf(a),y=gf(b);
		if(x!=y)f[x]=y,sz[y]+=sz[x];
	}
}es[207*207];
int gcd(int a,int b){
	return b?gcd(b,a%b):a;
}
int pw[1007],ans;
int main(){
cin >> n;
	pw[0]=1;
	for(int i=1;i<=n;++i)pw[i]=pw[i-1]*2%P;
	ans=(pw[n]-n-1)%P;
	for(int i=0;i<n;++i)scanf("%d%d",&ps[i].x,&ps[i].y);
	for(int i=0;i<n;++i){
		for(int j=0;j<i;++j){
			int x=ps[i].x-ps[j].x,y=ps[i].y-ps[j].y;
			int g=gcd(x,y);
			x/=g,y/=g;
			if(!x)y=1;
			if(!y)x=1;
			if(x<0)x=-x,y=-y;
			es[ep++]=(edge){i,j,x*30000+y};
		}
	}
	std::sort(es,es+ep);
	for(int i=0,j=0;i<ep;i=j){
		for(int a=0;a<n;++a)sz[f[a]=a]=1;
		for(;j<ep&&es[i].v==es[j].v;es[j++].cal());
		for(int a=0;a<n;++a)if(f[a]==a&&sz[a]>1)ans=(ans-pw[sz[a]]+sz[a]+1)%P;
	}
	cout << (ans % P + P ) % P;
	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>
using namespace std;
const int P=998244353;
int n;
struct pos{
	int x,y;
}ps[1007];
int ep=0,f[1007],sz[1007];
int gf(int x){while(x!=f[x])x=f[x]=x[f][f];return x;}
struct edge{
	int a,b;
	int v;
	bool operator<(const edge&e)const{return v<e.v;}
	void cal(){
		int x=gf(a),y=gf(b);
		if(x!=y)f[x]=y,sz[y]+=sz[x];
	}
}es[207*207];
int gcd(int a,int b){
	return b?gcd(b,a%b):a;
}
int pw[1007],ans;
int main(){
cin >> n;
	pw[0]=1;
	for(int i=1;i<=n;++i)pw[i]=pw[i-1]*2%P;
	ans=(pw[n]-n-1)%P;
	for(int i=0;i<n;++i)scanf("%d%d",&ps[i].x,&ps[i].y);
	for(int i=0;i<n;++i){
		for(int j=0;j<i;++j){
			int x=ps[i].x-ps[j].x,y=ps[i].y-ps[j].y;
			int g=gcd(x,y);
			x/=g,y/=g;
			if(!x)y=1;
			if(!y)x=1;
			if(x<0)x=-x,y=-y;
			es[ep++]=(edge){i,j,x*30000+y};
		}
	}
	std::sort(es,es+ep);
	for(int i=0,j=0;i<ep;i=j){
		for(int a=0;a<n;++a)sz[f[a]=a]=1;
		for(;j<ep&&es[i].v==es[j].v;es[j++].cal());
		for(int a=0;a<n;++a)if(f[a]==a&&sz[a]>1)ans=(ans-pw[sz[a]]+sz[a]+1)%P;
	}
	cout << (ans % P + P ) % P;
	return 0;
}
```