
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 discovered that among a group of identical-looking coins, one is counterfeit (lighter). He asked an expert to identify this coin using a balance scale without weights, and stipulated that each coin should participate in the weighings no more than twice. What is the largest 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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