
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
Grogg takes an a × b × c rectangular block (where a, b, c are positive integers), paints
the outside of it purple, and cuts it into abc small 1 × 1 × 1 cubes. He then places all the
small cubes into a bag, and Winnie reaches in and randomly picks one of the small cubes.
If the probability that Winnie picks a totally unpainted cube is 20%, determine all possible
values of the number of cubes in the bag.

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