#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
#define all(x) (x).begin(),(x).end()
const int mod=998244353,MAX=1005,INF=1<<30;

ll rui(ll a,ll b){
    ll ans=1;
    while(b>0){
        if(b&1) ans=ans*a%mod;
        a=a*a%mod;
        b/=2;
    }
    return ans;
}

int main(){
    
    int N;cin>>N;
    vector<int> X(N),Y(N);
    for(int i=0;i<N;i++){
        cin>>X[i]>>Y[i];
    }
    
    ll ans=rui(2,N);
    ans+=mod-(N+1);
    ans%=mod;
    
    for(int a=0;a<N;a++){
        for(int b=a+1;b<N;b++){
            ll cnt=0;
            int left=a,right=b;
            for(int k=0;k<N;k++){
                //if(k==a||k==b) continue;
                if((X[k]-X[a])*(Y[b]-Y[a])-(X[b]-X[a])*(Y[k]-Y[a])==0){
                    left=min(left,k);
                    right=max(right,k);
                    cnt++;
                }
            }
            if(left==a&&right==b) ans+=mod-(rui(2,cnt)-cnt-1)%mod;
            ans%=mod;
        }
    }
    
    cout<<ans<<endl;
}

### Prompt

Create a solution in cpp for 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;
#define all(x) (x).begin(),(x).end()
const int mod=998244353,MAX=1005,INF=1<<30;

ll rui(ll a,ll b){
    ll ans=1;
    while(b>0){
        if(b&1) ans=ans*a%mod;
        a=a*a%mod;
        b/=2;
    }
    return ans;
}

int main(){
    
    int N;cin>>N;
    vector<int> X(N),Y(N);
    for(int i=0;i<N;i++){
        cin>>X[i]>>Y[i];
    }
    
    ll ans=rui(2,N);
    ans+=mod-(N+1);
    ans%=mod;
    
    for(int a=0;a<N;a++){
        for(int b=a+1;b<N;b++){
            ll cnt=0;
            int left=a,right=b;
            for(int k=0;k<N;k++){
                //if(k==a||k==b) continue;
                if((X[k]-X[a])*(Y[b]-Y[a])-(X[b]-X[a])*(Y[k]-Y[a])==0){
                    left=min(left,k);
                    right=max(right,k);
                    cnt++;
                }
            }
            if(left==a&&right==b) ans+=mod-(rui(2,cnt)-cnt-1)%mod;
            ans%=mod;
        }
    }
    
    cout<<ans<<endl;
}

```