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After taking an Ancient Irish History class, Diarmuid has become interested in finding geometric patterns in neolithic tomb locations around Ireland. He carefully plots the locations of N tombs (2 ≤ N ≤ 1000), each one occupying a distinct point in the 2D plane, and he wonders how many different lines of symmetry exist for this set of points. A line of symmetry, of course, is a line across which the points on both sides are mirror images of each other.
Help Diarmuid answer this most pressing geometric question.
First line: The number of tombs N.
The following N lines: Two integers x, y separated by a space indicating the position x and y of each tomb (-10,000 ≤ x,y ≤ 10,000).
The number of different lines of symmetry of the point set.
3 0 0 0 1 0 3
4 0 0 0 1 1 0 1 1