#include<bits/stdc++.h>
using namespace std;

typedef long long ll;

template <typename T>
T gcd(T x,T y)
{
	while(y)swap(x%=y,y);
	return x;
}

#define rep(i,l,r) for(int i=l;i<=r;++i)

const int N=200+5,U=2e4+5,D=998244353;
int mi_2[N];
struct point
{
	int x,y;
};
point operator -(const point &a)
{
	return (point){-a.x,-a.y};
}
point operator -(const point &a,const point &b)
{
	return (point){a.x-b.x,a.y-b.y};
}
point operator /(const point &a,int k)
{
	return (point){a.x/k,a.y/k};
}
point standard(point p)
{
	if(p.x==0)return (point){0,1};
	if(p.x<0)p=-p;
	return p/gcd(p.x,abs(p.y));
}
int dy(const point &p)
{
	return p.x*U+p.y;
}
point p[N];
int F2(int n)
{
	return mi_2[n]-1-n;
}

int main()
{
	int n;
	cin>>n;
	rep(i,1,n)cin>>p[i].x>>p[i].y;
	mi_2[0]=1;
	rep(i,1,n)mi_2[i]=mi_2[i-1]*2%D;
	ll ans=mi_2[n]-1-n-n*(n-1)/2;
	rep(i,1,n)
	{
		unordered_map<int,int>mp;
		rep(j,i+1,n)++mp[dy(standard(p[j]-p[i]))];
		for(auto pr:mp)ans-=F2(pr.second);
	}
	cout<<(ans%D+D)%D<<endl;
}
### Prompt

Please formulate a cpp solution to the following problem:

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;

typedef long long ll;

template <typename T>
T gcd(T x,T y)
{
	while(y)swap(x%=y,y);
	return x;
}

#define rep(i,l,r) for(int i=l;i<=r;++i)

const int N=200+5,U=2e4+5,D=998244353;
int mi_2[N];
struct point
{
	int x,y;
};
point operator -(const point &a)
{
	return (point){-a.x,-a.y};
}
point operator -(const point &a,const point &b)
{
	return (point){a.x-b.x,a.y-b.y};
}
point operator /(const point &a,int k)
{
	return (point){a.x/k,a.y/k};
}
point standard(point p)
{
	if(p.x==0)return (point){0,1};
	if(p.x<0)p=-p;
	return p/gcd(p.x,abs(p.y));
}
int dy(const point &p)
{
	return p.x*U+p.y;
}
point p[N];
int F2(int n)
{
	return mi_2[n]-1-n;
}

int main()
{
	int n;
	cin>>n;
	rep(i,1,n)cin>>p[i].x>>p[i].y;
	mi_2[0]=1;
	rep(i,1,n)mi_2[i]=mi_2[i-1]*2%D;
	ll ans=mi_2[n]-1-n-n*(n-1)/2;
	rep(i,1,n)
	{
		unordered_map<int,int>mp;
		rep(j,i+1,n)++mp[dy(standard(p[j]-p[i]))];
		for(auto pr:mp)ans-=F2(pr.second);
	}
	cout<<(ans%D+D)%D<<endl;
}
```