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

const ll mod=998244353;
const int N=1005;
int n,x[N],y[N];

ll fpow(ll a,ll p){
	ll res=1;
	while(p){
		if(p&1) res=res*a%mod;
		a=a*a%mod;
		p>>=1;
	}
	return res;
}

int main() {
	ios_base::sync_with_stdio(0);cin.tie(0);
	cin>>n;
	for(int i=0;i<n;++i)
		cin>>x[i]>>y[i];
	ll res=fpow(2,n);
	res=(res-1+mod)%mod;
	res=(res-n+mod)%mod;
	res=(res-n*(n-1)/2+mod)%mod;
	for(int i=0;i<n;++i)
		for(int j=i+1;j<n;++j){
			int a=y[j]-y[i];
			int b=x[i]-x[j];
			int c=-a*x[i]-b*y[i];
			ll cnt=2;
			for(int k=0;k<n;++k)
				if(k!=i&&k!=j){
					int t=a*x[k]+b*y[k]+c;
					if(t!=0) continue;
					if(k<j) break;
					cnt++;
				}
			cnt%=mod;
			res=(res-fpow(2,cnt)+mod)%mod;
			res=(res+1)%mod;
			res=(res+cnt)%mod;
			res=(res+cnt*(cnt-1)/2)%mod;
		}
	cout<<res<<'\n';
}



### 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;
#define ll long long

const ll mod=998244353;
const int N=1005;
int n,x[N],y[N];

ll fpow(ll a,ll p){
	ll res=1;
	while(p){
		if(p&1) res=res*a%mod;
		a=a*a%mod;
		p>>=1;
	}
	return res;
}

int main() {
	ios_base::sync_with_stdio(0);cin.tie(0);
	cin>>n;
	for(int i=0;i<n;++i)
		cin>>x[i]>>y[i];
	ll res=fpow(2,n);
	res=(res-1+mod)%mod;
	res=(res-n+mod)%mod;
	res=(res-n*(n-1)/2+mod)%mod;
	for(int i=0;i<n;++i)
		for(int j=i+1;j<n;++j){
			int a=y[j]-y[i];
			int b=x[i]-x[j];
			int c=-a*x[i]-b*y[i];
			ll cnt=2;
			for(int k=0;k<n;++k)
				if(k!=i&&k!=j){
					int t=a*x[k]+b*y[k]+c;
					if(t!=0) continue;
					if(k<j) break;
					cnt++;
				}
			cnt%=mod;
			res=(res-fpow(2,cnt)+mod)%mod;
			res=(res+1)%mod;
			res=(res+cnt)%mod;
			res=(res+cnt*(cnt-1)/2)%mod;
		}
	cout<<res<<'\n';
}



```