We have $N$ points $P_1, P_2, \dots, P_N$ on a plane. The coordinates of $P_i$ is $(X_i, Y_i)$. We know that no three points lie on the same line.
For a subset $S$ of $\{ P_1, P_2, \dots, P_N \}$ with at least three elements, let us define the convex hull of $S$ as follows:

• The convex hull is the convex polygon with the minimum area such that every point of $S$ is within or on the circumference of that polygon.

Find the number, modulo $(10^9 + 7)$, of subsets SS such that the area of the convex hull is an integer.

Input is given from Standard Input in the following format:
$N$
$X_1\ Y_1$
$X_2\ Y_2$
$\vdots$
$X_N\ Y_N$

Print the answer. Note that you are asked to find the count modulo $(10^9 + 7)$.

$3≤N≤80$
$0 \leq |X_i|, |Y_i| \leq 10^4$
No three points lie on the same line.
All values in input are integers.

2

$\{P_1,P_2,P_4\}$ and $\{ P_2, P_3, P_4 \}$ satisfy the condition.