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#1*n*1#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

4 -1 0 0 2 1 0 0 -1 3 -666 -42 57 -84 19 282 3 -241 -50 307 43 -334 498 0

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

Local_UVa'2003