Problem A: Polylops 

Given the vertices of a non-degenerate polygon (no 180-degree angles, zero-length sides, or self-intersection - but not necessarily convex), you must determine how many distinct lines of symmetry exist for that polygon. A line of symmetry is one on which the polygon, when reflected on that line, maps to itself.


Input consists of a description of several polygons.

Each polygon description consists of two lines. The first contains the integer n ( 31#1n1#11000), which gives the number of vertices on the polygon. The second contains n pairs of numbers (an x- and a y-value), describing the vertices of the polygon in order. All coordinates are integers from -1000 to 1000.

Input terminates on a polygon with 0 vertices.


For every polygon described, print out a line saying `Polygon #x has y symmetry line(s).', where x is the number of the polygon (starting from 1), and y is the number of distinct symmetry lines on that polygon.

Sample Input 

-1 0 0 2 1 0 0 -1
-666 -42 57 -84 19 282
-241 -50 307 43 -334 498

Sample Output 

Polygon #1 has 1 symmetry line(s).
Polygon #2 has 0 symmetry line(s).
Polygon #3 has 1 symmetry line(s).