IN-TST 2003 Problem 6

A zig-zag in the plane consists of two parallel half-lines connected by a line segment. Find $z_n$ , the maximum number of regions into which $n$ zig-zags can divide the plane. For example, $z_1=2,z_2=12$ (see the diagram). Of these $z_n$ regions how many are bounded? [The zig-zags can be as narrow as you please.] Express your answers as polynomials in $n$ of degree not exceeding $2$ .