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::io::{Read, BufReader};

macro_rules! read {
    ( $( $t:ty ),* ) => {
        {
            let mut buf = String::new();
            io::stdin().read_line(&mut buf);
            let mut sp = buf.trim().split(" ");
            ($(
                {
                    sp.next().unwrap().parse::<$t>().unwrap()
                },
            )*)
        }
    }
}
macro_rules! read_vec {
    ($t:ty) => {
        {
            let mut buf = String::new();
            io::stdin().read_line(&mut buf);
            let mut sp = buf.trim().split(" ");
            let mut ret = vec![];
            for s in sp {
                ret.push(s.parse::<$t>().unwrap());
            }
            ret
        }
    }
}

const MOD: i64 = 998244353;

fn line(a: (i64, i64), b: (i64, i64), c: (i64, i64)) -> bool {
    let b = (b.0 - a.0, b.1 - a.1);
    let c = (c.0 - a.0, c.1 - a.1);

    b.1 * c.0 == b.0 * c.1
}

fn ctri(ni: i64) -> i64 {
    let mut ret = 1i64;
    for _ in 0..ni {
        ret = ret * 2 % MOD;
    }

    ret = (ret + MOD - 1) % MOD;
    ret = (ret + MOD - ni) % MOD;
    ret = (ret + MOD - ni * (ni - 1) / 2) % MOD;

    ret
}

fn main() {
    let (n,) = read!(usize);
    let mut pts = vec![];
    for _ in 0..n {
        pts.push(read!(i64, i64));
    }
    
    let mut lin = vec![0i64; n + 1];

    for i in 0..n {
        for j in (i + 1)..n {
            let mut cnt = 0;
            for k in 0..n {
                if line(pts[i], pts[j], pts[k]) {
                    cnt += 1
                }
            }

            lin[cnt] += 1
        }
    }

    let mut ret = 1i64;
    for _ in 0..n {
        ret = ret * 2 % MOD;
    }

    let ni = n as i64;
    ret = ctri(ni);

    for i in 3..(n + 1) {
        let act = lin[i] / ((i * (i - 1) / 2) as i64);
        ret = (ret + MOD - ctri(i as i64) * act % MOD) % MOD;
    }
    println!("{}", ret);
}