F. Symmetric Projections
geometry
You are given a set of n points on the plane. A line containing the origin is called good, if projection of the given set to this line forms a symmetric multiset of points. Find the total number of good lines.Multiset is a set where equal elements are allowed.Multiset is called symmetric, if there is a point P on the plane such that the multiset is centrally symmetric in respect of point P.
The first line contains a single integer n (1 ≤ n ≤ 2000) — the number of points in the set.Each of the next n lines contains two integers xi and yi ( - 106 ≤ xi, yi ≤ 106) — the coordinates of the points. It is guaranteed that no two points coincide.
If there are infinitely many good lines, print -1.Otherwise, print single integer — the number of good lines.
Picture to the first sample test: In the second sample, any line containing the origin is good.