Example 1. Use 5 different colors to color the 5 parts A, B, C, D, E in the figure. Adjacent parts cannot be the same color, but the same color can be reused or not used at all. How many different coloring methods that meet the requirements are there?
Analysis: First, color A with 5 options, then color B with 4 options since B cannot be the same color as A, next color C with 3 options since C cannot be the same color as A or B, D cannot be the same color as B or C, so there are 3 options, E cannot be the same color as C or D, so there are 3 options. In total, there are $5 \times 4 \times$ $3 \times 3 \times 3=540$ coloring methods.