
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
Suppose you have a  $6$  sided dice with  $3$  faces colored red,  $2$  faces colored blue, and  $1$  face colored green. You roll this dice  $20$  times and record the color that shows up on top. What is the expected value of the product of the number of red faces, blue faces, and green faces?

*Proposed by Daniel Li*

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