
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
In a circus, there are \( n \) clowns who dress and paint themselves up using a selection of 12 distinct colors. Each clown is required to use at least five different colors. The ringmaster orders that no two clowns have exactly the same set of colors and no more than 20 clowns may use any one particular color. Find the largest number \( n \) of clowns so as to make the ringmaster's orders possible.

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