Polylops

Time Limit: 2 Seconds
Memory Limit: 65536 KB

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**

Input consists of a description of several polygons.

Each polygon description consists of two lines. The first contains the integer
"n" (3 <= n <= 1000), 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.

**Output**

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**

4

-1 0 0 2 1 0 0 -1

3

-666 -42 57 -84 19 282

3

-241 -50 307 43 -334 498

0

**Sample Output**

Polygon #1 has 1 symmetry line(s).

Polygon #2 has 0 symmetry line(s).

Polygon #3 has 1 symmetry line(s).

Source:

**University of Waterloo Local Contest 1998.10.04**
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