AUSTRIAN-POLISH 1983 Problem 9

To each side of the regular $p$ -gon of side length $1$ there is attached a $1 \times k$ rectangle, partitioned into $k$ unit cells, where $k$ and $p$ are given positive integers and p an odd prime. Let $P$ be the resulting nonconvex star-like polygonal figure consisting of $kp + 1$ regions ( $kp$ unit cells and the $p$ -gon). Each region is to be colored in one of three colors, adjacent regions having different colors. Furthermore, it is required that the colored figure should not have a symmetry axis. In how many ways can this be done?