Axes of Symmetry

Little Johnny, a gifted young mathematician, has a younger sister named Justina. Johnny is happy to help her with homework, but he tires of checking the same problems over and over. One day the math teacher assigned many exercises about finding the axes of symmetry of various figures, and Justina wants to double-check her answers many times. Since Johnny would rather leave for a trip to the seaside, he asks you to write a program that checks Justina's answers for him.

Write a program that:

• reads the descriptions of several polygons from standard input,
• determines the number of axes of symmetry of each polygon,
• writes the results to standard output.

An axis of symmetry is a line such that folding the figure along it makes the figure coincide exactly with itself.

The first line contains the number of polygons $t$ ($1 \le t \le 10$). Then $t$ polygon descriptions follow.

The first line of each description contains the number of vertices $n$ ($3 \le n \le 10^5$). Each of the next $n$ lines contains two integers $x$ and $y$ ($-10^8 \le x, y \le 10^8$), the coordinates of the vertices given in order along the polygon.

The polygons need not be convex, but they have no self-intersections: any two sides share at most one point (their common endpoint, if they actually touch). Moreover, no two consecutive sides are parallel.

Output exactly $t$ lines. The $k$-th line contains a single integer $n_k$, the number of axes of symmetry of the $k$-th polygon.

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