ConvexScore

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°.

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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.

Examples

Input

4 0 0 0 1 1 0 1 1

Output

5

Input

5 0 0 0 1 0 2 0 3 1 1

Output

11

Input

1 3141 2718

Output

0

inputFormat

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

outputFormat

Output

Print the sum of all the scores modulo 998244353.

Examples

Input

4 0 0 0 1 1 0 1 1

Output

5

Input

5 0 0 0 1 0 2 0 3 1 1

Output

11

Input

1 3141 2718

Output

0

样例

1
3141 2718

0