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: Time Limit Exceeded)

n=gets.to_i
a=[]
n.times do
  a<< gets.split.map(&:to_i)
end
b=Array.new(n).map{[]}
r=[]
x=[]
0.upto(n-3) do |i|
  (i+1).upto(n-2) do |j|
    g = 1.0*(a[i][0]-a[j][0])/(a[i][1]-a[j][1])
    g=10**9 if a[i][1]-a[j][1] == 0
    t=0
    (j+1).upto(n-1) do |k|
      c=false
      2.times do |l|
        if a[i][l]==a[j][l] && a[i][l]==a[k][l]
          c=true
        end
      end
      gg = 1.0*(a[i][0]-a[k][0])/(a[i][1]-a[k][1])
      gg = 10**9 if a[i][1]-a[k][1] == 0
      if g == gg
        c=true
      end
      if c
        x<< [j,k]
        if !x.include?([i,k])
          t+=1
          x<< [i,k]
        end
      end
    end
    r<< t+2 if t!=0
  end
end
y=[1,1]
2.upto(200) do |i|
  y<< y[-1]*i
end
$y=y
tt=0
def ncm n,m
  $y[n]/$y[n-m]/$y[m]
end
if n>=3
  tt+=(g=ncm n,3)
  (n-3).times do |i|
    tt+=(g=g*(n-3-i)/(4+i))
  end
  r.each do |e|
    tt-=(g=ncm e,3)
    (e-3).times do |i|
      tt-=(g=g*(e-3-i)/(4+i))
    end
  end
end
p tt%998244353