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 Ruby (2.3.3) (judge verdict: Wrong Answer)

N = gets.to_i
xs, ys = [], []
N.times do |i|
    xs[i], ys[i] = gets.split.map(&:to_i)
end
MOD = 998244353

pmod2 = [1]
(1..N).each do |i|
    pmod2[i] = pmod2[i-1] * 2 % MOD
end

ans = (pmod2[N] - N - 1) % MOD
checked = Array.new(N){Array.new(N)}

N.times do |i|
    ((i+1)...N).each do |j|
        next if checked[i][j]
        pts = [i, j]
        ((j+1)...N).each do |k|
            if (xs[j]-xs[i]) * (ys[k]-ys[i]) == (xs[k]-xs[i]) * (ys[j]-ys[i])
                pts.push(k)
            end
        end
        num = pts.length
        ans = (ans - pmod2[num] + num + 1) % MOD
        num.times do |a|
            ((a+1)...num).each do |b|
                checked[a][b] = true
            end
        end
    end
end
puts ans