<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 There are 4 different colors of bulbs (each available in sufficient quantity). We need to install a bulb at each vertex of a triangular prism \(ABC-A_1B_1C_1\) such that the bulbs at the endpoints of each edge have different colors. Additionally, each color must be used at least once. How many distinct ways are there to install the bulbs? Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
432
There are 4 different colors of bulbs (each available in sufficient quantity). We need to install a bulb at each vertex of a triangular prism \(ABC-A_1B_1C_1\) such that the bulbs at the endpoints of each edge have different colors. Additionally, each color must be used at least once. How many distinct ways are there to install the bulbs?
test