Quadrilaterals

A quadrilateral is a polygon with exactly four distinct corners and four distinct sides, with no crossing between its sides. A quadrilateral is called convex if all the inner angles at its corners are less than 180 degrees. Otherwise it is non-convex. The illustration below shows a convex quadrilateral (left) and a non-convex quadrilateral (right).

In a test problem, you are given a set $P$ of $n$ points in the plane. No three points in $P$ are collinear. Your task is to find as many quadrilaterals as possible by connecting four points from $P$. Each point in $P$ can be used any number of times, and the quadrilaterals you find may overlap each other. You earn credits for each quadrilateral based on its shape and area. Convex quadrilaterals and quadrilaterals with minimum area earn more.

Let $a$ be the minimum area over all quadrilaterals that can be formed by connecting four points in $P$. The credits are as follows.

• Each distinct convex quadrilateral with area exactly $a$ earns 4 credits.
• Each distinct non-convex quadrilateral with area exactly $a$ earns 3 credits.
• Each distinct convex quadrilateral with area strictly larger than $a$ earns 2 credits.
• Each distinct non-convex quadrilateral with area strictly larger than $a$ earns 1 credit.

Two quadrilaterals are distinct unless their corners and sides are exactly the same. There may be two or more quadrilaterals with the minimum area $a$.

Given $P$, write a program that outputs the maximum total credits you can earn by finding all possible quadrilaterals from $P$.

The first line contains an integer $n$ ($4 \le n \le 1000$), the number of points in $P$. Each of the next $n$ lines contains two integers, the x and y coordinates of a point in $P$. Each coordinate is between $-10^{9}$ and $10^{9}$, inclusive. No three points in $P$ are collinear.

Print one line containing a single integer, the maximum total credits you can earn from $P$.