
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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?

Present the answer in LaTex format: \boxed{Your answer}
user