P3454 [POI 2007] OSI-Axes of Symmetry

Little Johnny - a well-respected young mathematician - has a younger sister, Justina. Johnny likes hissister very much and he gladly helps her with her homework, but, like most scientific minds, he does mindsolving the same problems again. Unfortunately, Justina is a very diligent pupil, and so she asks Johnny toreview her assignments many times, for sake of certainty. One sunny Friday, just before the famous LongMay Weekend1 the math teacher gave many exercises consisting in finding the axes of symmetry of variousgeometric figures. Justina is most likely to spend considerable amount of time solving these tasks. LittleJohnny had arranged himself a trip to the seaside long time before, nevertheless he feels obliged to help hislittle sister. Soon, he has found a solution - it would be best to write a programme that wouldease checking Justina's solutions. Since Johnny is a mathematician, not a computer scientist, and you are hisbest friend, it falls to you to write it. ## Task Write a programme that: - reads the descriptions of the polygons from the standard input, - determines the number of axes of symmetry for each one of them, - writes the result to the standard output. 给定一个多边形，求对称轴数量。

In the first line of the input there is one integer $t$ ($1 \le t \le 10$) - it is the number of polygons, for which the number of axes of symmetry is to be determined. Next, $t$ descriptions of the polygons follow. The first line of each description contains one integer $n$ ($3 \le n \le 100\ 000$) denoting the number of vertices of the polygon. In each of the following $n$ lines there are two integers $x$ and $y$ ($-100\ 000\ 000 \le x, y \le 100\ 000\ 000$) representing the coordinates of subsequent vertices of the polygon. The polygons need not be convex, but they have no self-intersections - any two sides have at most one common point - their common endpoint, if they actually share it. Furthermore, no pair of consecutive sides is parallel.

Your programme should output exactly $t$ lines, with the $k$'th line containing a sole integer $n_k$ - the number of axes of symmetry of the $k$'th polygon.