#include<bits/stdc++.h>
using namespace std;
typedef long long int uli;
const int mx=222;
const uli mod=998244353;
uli pwr[mx];
int x[mx],y[mx];
int gcd(int a,int b){ 
  if(b==0)return a;
  return gcd(b,a%b);
}
bool inside(int k,int i,int j){
  if(k==i || k==j)return false;
  int a=x[k]-x[i],b=y[k]-y[i];
  int g=gcd(abs(a),abs(b));
  a/=g,b/=g;
  int c=x[j]-x[k],d=y[j]-y[k];
  g=gcd(abs(c),abs(d));
  c/=g,d/=g;
  return a==c && b==d;
}
int main(){
  pwr[0]=1;
  for(int i=1;i<mx;i++)pwr[i]=pwr[i-1]*2ll%mod;
  int n;
  scanf("%d",&n);
  for(int i=0;i<n;i++)
    scanf("%d %d",x+i,y+i);
  uli ans=pwr[n];
  for(int i=0;i<n;i++){ 
    for(int j=i+1;j<n;j++){
      int cnt=0;
      for(int k=0;k<n;k++)
        if(inside(k,i,j))
          cnt++;
      ans=(ans-pwr[cnt]+mod)%mod;
    }
  }
  ans=(ans-n-1+mod)%mod;
  printf("%lld\n",ans);
  return 0;
}

### Prompt

Your task is to create 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 int uli;
const int mx=222;
const uli mod=998244353;
uli pwr[mx];
int x[mx],y[mx];
int gcd(int a,int b){ 
  if(b==0)return a;
  return gcd(b,a%b);
}
bool inside(int k,int i,int j){
  if(k==i || k==j)return false;
  int a=x[k]-x[i],b=y[k]-y[i];
  int g=gcd(abs(a),abs(b));
  a/=g,b/=g;
  int c=x[j]-x[k],d=y[j]-y[k];
  g=gcd(abs(c),abs(d));
  c/=g,d/=g;
  return a==c && b==d;
}
int main(){
  pwr[0]=1;
  for(int i=1;i<mx;i++)pwr[i]=pwr[i-1]*2ll%mod;
  int n;
  scanf("%d",&n);
  for(int i=0;i<n;i++)
    scanf("%d %d",x+i,y+i);
  uli ans=pwr[n];
  for(int i=0;i<n;i++){ 
    for(int j=i+1;j<n;j++){
      int cnt=0;
      for(int k=0;k<n;k++)
        if(inside(k,i,j))
          cnt++;
      ans=(ans-pwr[cnt]+mod)%mod;
    }
  }
  ans=(ans-n-1+mod)%mod;
  printf("%lld\n",ans);
  return 0;
}

```