
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
A banker found out that among a set of identical-looking coins, one is counterfeit (lighter in weight). He asked an expert to identify the counterfeit coin using a balance scale but stipulated that each coin can be used in weighings no more than twice. What is the maximum number of coins the banker can have so that the expert can definitely identify the counterfeit coin in $n$ weighings?

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