Symmetry

Farmer John has plotted the locations of his $N$ cows ($2 \le N \le 1000$) on the 2D plane, each cow occupying a distinct point. He wonders how many distinct lines of symmetry this set of points has.

A line of symmetry is a line such that reflecting every point across it yields exactly the same set of points; in other words, the points on the two sides of the line are mirror images of each other.

Count the number of lines of symmetry of the point set.

• Line $1$: the single integer $N$.
• Lines $2 \dots N+1$: line $i+1$ contains two space-separated integers, the $x$ and $y$ coordinates of the $i$-th cow ($-10000 \le x, y \le 10000$).

• Line $1$: the number of distinct lines of symmetry of the point set.

For example, four points forming the corners of a square have $4$ lines of symmetry: one vertical, one horizontal, and two diagonal.