#include <bits/stdc++.h>
using namespace std;
#define FOR(i,a,b) for(int i=(a);i<(b);++i)
#define rep(i,n)   FOR(i,0,n)
#define pb emplace_back
typedef long long ll;
typedef pair<int,int> pint;

const int mod=998244353;
ll p[2001];
int x[201],y[201];
bool used[201][201];
int main(){
    int n;
    cin>>n;
    rep(i,n){
        cin>>x[i]>>y[i];
    }
    p[0]=1;
    rep(i,2000) p[i+1]=p[i]*2%mod;
    ll sum=n+1;
    rep(i,n)FOR(j,i+1,n){
        if(used[i][j]) continue;
        int cnt=0;
        vector<int> u;
        rep(k,n){
            int a=(x[k]-x[i])*(y[j]-y[i]);
            int b=(x[j]-x[i])*(y[k]-y[i]);
            if(a==b){
                //used[i][j]=used[j][i]=true;
                u.pb(k);
                ++cnt;
            } 
        }
        for(int it1:u)for(int it2:u) used[it1][it2]=true;
        sum+=p[cnt]-cnt-1;
        sum%=mod;
    }
    ll ans=(p[n]-sum+mod)%mod;
    cout<<ans<<endl;
    return 0;
}
### Prompt

Generate 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 FOR(i,a,b) for(int i=(a);i<(b);++i)
#define rep(i,n)   FOR(i,0,n)
#define pb emplace_back
typedef long long ll;
typedef pair<int,int> pint;

const int mod=998244353;
ll p[2001];
int x[201],y[201];
bool used[201][201];
int main(){
    int n;
    cin>>n;
    rep(i,n){
        cin>>x[i]>>y[i];
    }
    p[0]=1;
    rep(i,2000) p[i+1]=p[i]*2%mod;
    ll sum=n+1;
    rep(i,n)FOR(j,i+1,n){
        if(used[i][j]) continue;
        int cnt=0;
        vector<int> u;
        rep(k,n){
            int a=(x[k]-x[i])*(y[j]-y[i]);
            int b=(x[j]-x[i])*(y[k]-y[i]);
            if(a==b){
                //used[i][j]=used[j][i]=true;
                u.pb(k);
                ++cnt;
            } 
        }
        for(int it1:u)for(int it2:u) used[it1][it2]=true;
        sum+=p[cnt]-cnt-1;
        sum%=mod;
    }
    ll ans=(p[n]-sum+mod)%mod;
    cout<<ans<<endl;
    return 0;
}
```