E - ConvexScore

Time Limit: 2 sec / Memory Limit: 256 MB

問題文

S に対し、N 個の点のうち S の凸包の内部と境界 (頂点も含む) に含まれる点の個数を n とおき、 S のスコアを、2^{n-|S|} と定義します。

ただし答えはとても大きくなりうるので、998244353 で割った余りをかわりに求めてください。

制約

• 1≤N≤200
• 0≤x_i,y_i<10^4 (1≤i≤N)
• i≠j なら x_i≠x_j または y_i≠y_j
• x_i,y_i は整数

入力

N
x_1 y_1
x_2 y_2
:
x_N y_N


出力

スコアの総和を 998244353 で割った余りを出力せよ。

入力例 1

4
0 0
0 1
1 0
1 1


出力例 1

5


S として三角形（をなす点集合）が 4 つと四角形が 1 つとれます。どれもスコアは 2^0=1 となるので、答えは 5 です。

入力例 2

5
0 0
0 1
0 2
0 3
1 1


出力例 2

11


スコア 1 の三角形が 3 つ、スコア 2 の三角形が2つ、スコア 4 の三角形が 1 つあるので、答えは 11 です。

入力例 3

1
3141 2718


出力例 3

0


S として考えられるものがないので、答えは 0 です。

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.