Example 5 (2008 Chongqing College Entrance Examination Question) A person has 4 different colored light bulbs (with enough of each color), to be installed on the 6 points $A, B, C, A_{1}, B_{1}, C_{1}$ as shown in Figure 5, with the requirement that the light bulbs at the ends of the same line segment must be of different colors. The number of installation methods where each color of light bulb is used at least once is $\qquad$ (answer with a number).