
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 four cups, each with the mouth facing up. Each time, three cups are flipped, and cups that have been flipped before are allowed to be flipped again. After $n$ flips, all the cups have the mouth facing down. What is the smallest value of the positive integer $n$?

Note: Flipping a cup means turning it from mouth up to mouth down or from mouth down to mouth up.

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