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::FromStr;
use std::collections::HashMap;

#[allow(dead_code)]
fn read<T: FromStr>() -> T {
    let stdin = stdin();
    let stdin = stdin.lock();
    let token: String = stdin
        .bytes()
        .map(|c| c.expect("failed to read char") as char)
        .skip_while(|c| c.is_whitespace())
        .take_while(|c| !c.is_whitespace())
        .collect();
    token.parse().ok().expect("failed to parse token")
}

#[allow(dead_code)]
#[derive(Clone, Copy, Debug, PartialEq, PartialOrd, Hash)]
struct Point {
    x: i64,
    y: i64,
}
#[allow(dead_code)]
type Polygon = Vec<Point>;

impl std::cmp::Eq for Point {}
impl std::cmp::Ord for Point {
    fn cmp(&self, rhs: &Point) -> std::cmp::Ordering {
        (self.x, self.y).partial_cmp(&(rhs.x, rhs.y)).unwrap()
    }
}
impl std::ops::Add for Point {
    type Output = Point;
    fn add(self, rhs: Point) -> Point {
        Point {
            x: self.x + rhs.x,
            y: self.y + rhs.y,
        }
    }
}
impl std::ops::AddAssign for Point {
    fn add_assign(&mut self, rhs: Point) {
        self.x += rhs.x;
        self.y += rhs.y;
    }
}
impl std::ops::Sub for Point {
    type Output = Point;
    fn sub(self, rhs: Point) -> Point {
        Point {
            x: self.x - rhs.x,
            y: self.y - rhs.y,
        }
    }
}
impl std::ops::SubAssign for Point {
    fn sub_assign(&mut self, rhs: Point) {
        self.x -= rhs.x;
        self.y -= rhs.y;
    }
}
impl std::ops::Mul<Point> for i64 {
    type Output = Point;
    fn mul(self, rhs: Point) -> Point {
        Point {
            x: self * rhs.x,
            y: self * rhs.y,
        }
    }
}
impl std::ops::Mul<i64> for Point {
    type Output = Point;
    fn mul(self, rhs: i64) -> Point {
        Point {
            x: self.x * rhs,
            y: self.y * rhs,
        }
    }
}
impl std::ops::Div<i64> for Point {
    type Output = Point;
    fn div(self, rhs: i64) -> Point {
        Point {
            x: self.x / rhs,
            y: self.y / rhs,
        }
    }
}
#[allow(dead_code)]
impl Point {
    fn new(x: i64, y: i64) -> Point {
        Point { x: x, y: y }
    }
}

fn gcd(a: i64, b: i64) -> i64 {
    if b == 0 {
        return a;
    }
    gcd(b, a % b)
}

#[allow(dead_code)]
fn pow_mod(base: i64, power: i64, modulo: i64) -> i64 {
    let mut base = base;
    let mut power = power;
    let mut ans = 1;
    while power > 0 {
        if power & 1 == 1 {
            ans = ans * base % modulo;
        }
        power >>= 1;
        base = base * base % modulo;
    }
    ans
}

const MOD: i64 = 998244353;
fn main() {
    let n = read::<usize>();
    let mut ps = vec![Point::new(0i64, 0i64); n];
    for i in 0..n {
        let x = read::<i64>();
        let y = read::<i64>();
        ps[i] = Point::new(x, y);
    }
    let mut starts = HashMap::<((i64, i64), Point), usize>::new();
    let mut cnts = HashMap::<((i64, i64), Point), i64>::new();
    for i in 0..n {
        for j in i + 1..n {
            let mut vect = ps[j] - ps[i];
            let g = gcd(vect.x.abs(), vect.y.abs());
            vect = vect / g;
            if vect.x == 0 {
                vect = Point::new(0, 1);
            } else if vect.x < 0 {
                vect = vect * -1;
            }
            let mut yupper = ps[i].y * vect.x - ps[i].x * vect.y;
            let mut ylower = vect.x;
            if ylower != 0 {
                let g = gcd(yupper.abs(), ylower.abs());
                yupper /= g;
                ylower /= g;
            }
            // println!("{} {} {:?} {}", i, j, vect, g);
            let key = ((yupper, ylower), vect);
            if !starts.contains_key(&key) {
                starts.insert(key.clone(), i);
                cnts.insert(key.clone(), 2);
            } else if starts[&key] == i {
                *cnts.get_mut(&key).unwrap() += 1;
            }
        }
    }
    let mut ans = (pow_mod(2, n as i64, MOD) - 1 - n as i64 + MOD) % MOD;
    for &cnt in cnts.values() {
        // println!("{}", cnt);
        ans = (ans - pow_mod(2, cnt, MOD) + cnt + 1 + MOD) % MOD;
    }
    println!("{}", ans);
}