Example 6 Color each vertex of a quadrilateral pyramid with one color, and make the endpoints of the same edge have different colors. If only 5 colors are available, the total number of different coloring methods is $\qquad$ .
Solution 1 According to the problem, the vertices $S, A, B$ of the quadrilateral pyramid $S-ABCD$ must be of different colors, and there are $P_{5}^{3}=60$ ways to color them.

When the colors of $S, A, B$ are determined, let's assume their colors are 1, 2, and 3, respectively. Then $C$ can only be colored with one of the colors 2, 4, or 5.
(1) If $C$ is colored 2, then $D$ can be colored with one of the colors 3, 4, or 5, which gives $C_{3}^{1}=3$ ways.
(2) If $C$ is colored 4, then $D$ can be colored with one of the colors 3 or 5, which gives $C_{2}^{1}=2$ ways.
(3) If $C$ is colored 5, then $D$ can be colored with one of the colors 2 or 4, which gives $C_{2}^{1}=2$ ways.
Therefore, the total number of coloring methods is $60(3+2+2)=420$.