Problem Statement

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 S such that the area of the convex hull is an integer.

Input

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

Output

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

Constraints

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

Sample Input 1

4
0 0
1 2
0 1
1 0

Sample Output 1

2

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

Sample Input 2

23
-5255 7890
5823 7526
5485 -113
7302 5708
9149 2722
4904 -3918
8566 -3267
-3759 2474
-7286 -1043
-1230 1780
3377 -7044
-2596 -6003
5813 -9452
-9889 -7423
2377 1811
5351 4551
-1354 -9611
4244 1958
8864 -9889
507 -8923
6948 -5016
-6139 2769
4103 9241

Sample Output 2

4060436

见题面 Input 部分

见题面 Output 部分