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 Java8 (OpenJDK 1.8.0) (judge verdict: Runtime Error)

import java.io.FileNotFoundException;
import java.io.PrintWriter;
import java.util.Arrays;
import java.util.PriorityQueue;
import java.util.Random;
import java.util.Scanner;

public class Main {
	public static void main(String[] args) throws FileNotFoundException {
		new Main().run();
	}

	final long MODULO = 998244353L;
	long[] fac = new long[400];
	long[] invfac = new long[400];
	long[] inv = new long[400];
	long[][] C = new long[400][400];
	long[] pow2 = new long[400];
	{
		fac[0] = 1;
		invfac[0] = 1;
		inv[1] = 1;
		for (int i = 1; i < fac.length; ++i) {
			fac[i] = fac[i - 1] * i % MODULO;
		}
		for (int i = 2; i < inv.length; ++i) {
			inv[i] = (MODULO - (MODULO / i) * inv[(int) (MODULO % i)] % MODULO) % MODULO;
		}
		for (int i = 1; i < invfac.length; ++i) {
			invfac[i] = invfac[i - 1] * inv[i] % MODULO;
		}
		C[0][0] = 1;
		for (int i = 1; i < C.length; ++i) {
			for (int j = 0; j <= i; ++j) {
				C[i][j] = (j > 0 ? C[i - 1][j - 1] : 0) + C[i - 1][j];
				C[i][j] %= MODULO;
			}
		}
		pow2[0] = 1;
		for (int i = 1; i < pow2.length; ++i) {
			pow2[i] = pow2[i - 1] * 2 % MODULO;
		}

	}

	void run() {
		Scanner sc = new Scanner(System.in);
		int n = sc.nextInt();
		int[] x = new int[n];
		int[] y = new int[n];
		for (int i = 0; i < n; ++i) {
			x[i] = sc.nextInt();
			y[i] = sc.nextInt();
		}
		solver(n, x, y);
	}

	void solver(int n, int[] x, int[] y) {
		long ans = pow2[n];

		ans = (ans - n + MODULO) % MODULO;
		for (int i = 0; i < n; ++i) {
			boolean[] vis = new boolean[n];
			vis[i] = true;
			for (int j = i + 1; j < n; ++j) {
				if (vis[j])
					continue;
				vis[j] = true;
				int cnt = 1;
				for (int k = j + 1; k < n; ++k) {
					if (vis[k]) {
						throw new AssertionError();
					}
					if ((x[i] - x[j]) * (y[k] - y[i]) == (y[i] - y[j]) * (x[k] - x[i])) {
						vis[k] = true;
						++cnt;
					}
				}
				ans = (ans - (pow2[cnt] - 1) + MODULO) % MODULO;
			}
		}
		System.out.println(ans - 1);
	}

	static void tr(Object... objects) {
		System.out.println(Arrays.deepToString(objects));
	}
}