Coloring the Regions of a Ring-Shaped Flower Bed

A ring-shaped flower bed is divided into $5$ regions, as shown in the figure. Each region is to be planted with one kind of flower, chosen from $4$ available colors, and any two adjacent regions must receive different colors.

(1) Find the number of different planting schemes.

(2) Replace the number of regions in part (1) by $n$ (where $n \ge 3$), keeping the same rules. Find the number of different planting schemes.

(3) In part (2), replace the number of available colors by $m$ (where $m \ge 2$). Find the number of different planting schemes.