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 Rust (1.15.1) (judge verdict: Accepted)

use std::io;
use std::str;
use std::usize;
use std::cmp;

fn mod_pow(x: i64, mut exp: i64, modulo: i64) -> i64 {
    let mut result = 1;
    let mut cur = x;
    while exp > 0 {
        if exp & 1 == 1 {
            result = (result * cur) % modulo;
        }
        cur = (cur * cur) % modulo;
        exp >>= 1;
    }
    result
}

fn main() {
    let modulo = 998244353;
    let n = read_values::<usize>()[0];
    let (x, y) = {
        let mut x = vec![0; n];
        let mut y = vec![0; n];
        for i in 0..n {
            let v = read_values::<i64>();
            x[i] = v[0];
            y[i] = v[1];
        }
        (x, y)
    };

    let mut ans = mod_pow(2, n as i64, modulo);
    ans -= 1;
    ans -= n as i64;
    ans = (ans + modulo) % modulo;

    // 両端がそれぞれ i, j となるような線分に乗っている点の数を数える
    for i in 0..n {
        for j in (i + 1)..n {
            let dx = x[i] - x[j];
            let dy = y[i] - y[j];

            let mut count = 0;
            let mut left = i;
            let mut right = j;
            for k in 0..n {
                let kx = x[i] - x[k];
                let ky = y[i] - y[k];
                if dx * ky - kx * dy == 0 {
                    left = cmp::min(left, k);
                    right = cmp::max(right, k);
                    count += 1;
                }
            }

            // 両端が i, j でない時はスルー
            if left != i || right != j {
                continue;
            }

            // 凸包の面積が 0 であるような部分集合のサイズ
            let subset = mod_pow(2, count, modulo) - count - 1;
            ans = (ans + modulo - subset) % modulo;
        }
    }

    println!("{}", ans);
}

fn read_line() -> String {
    let stdin = io::stdin();
    let mut buf = String::new();
    stdin.read_line(&mut buf).unwrap();
    buf
}

fn read_values<T>() -> Vec<T>
    where T: std::str::FromStr,
          T::Err: std::fmt::Debug
{
    read_line()
        .split(' ')
        .map(|a| a.trim().parse().unwrap())
        .collect()
}