Problem:
Score : 700 points

Problem Statement

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°.

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.

Sample Input 1

4
0 0
0 1
1 0
1 1

Sample Output 1

5

We have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.

Sample Input 2

5
0 0
0 1
0 2
0 3
1 1

Sample Output 2

11

We have three "triangles" with a score of 1 each, two "triangles" with a score of 2 each, and one "triangle" with a score of 4. Thus, the answer is 11.

Sample Input 3

1
3141 2718

Sample Output 3

0

There are no possible set as S, so the answer is 0.

Solution in C++14 (GCC 5.4.1) (judge verdict: Accepted)

#include<cstdio>
#include<cstring>
#include<algorithm>
#define mod 998244353
#define MN 205
using namespace std;
inline int in(){
    int x=0;bool f=0; char c;
    for (;(c=getchar())<'0'||c>'9';f=c=='-');
    for (x=c-'0';(c=getchar())>='0'&&c<='9';x=(x<<3)+(x<<1)+c-'0');
    return f?-x:x;
}
struct st{
	int x,y;
}a[MN];
int pw[MN],n,sum,res;
inline bool cmp(st a,st b){
	return (a.x==b.x)?a.y<b.y:a.x<b.x; 
}
int main()
{
	n=in();pw[0]=1;
	for (int i=1;i<=n;++i) pw[i]=(pw[i-1]<<1)%mod; 
	for (int i=1;i<=n;++i) a[i].x=in(),a[i].y=in();
	sort(a+1,a+n+1,cmp);
	for (int i=1;i<=n;++i)
	for (int j=i+1;j<=n;++j){
		int cnt=0,tmp;
		for (int k=j+1;k<=n;++k)
		if ((1ll*(a[k].x-a[i].x)*(a[j].y-a[i].y))==(1ll*(a[j].x-a[i].x)*(a[k].y-a[i].y))) ++cnt;
		res=(res+pw[cnt])%mod;
	}
	sum=(pw[n]+mod-n-1)%mod;sum=(sum+mod-res)%mod;
	printf("%d",sum);return 0;
}