
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
We have $3^{2k}$ identical-looking coins, one of which is counterfeit and weighs slightly less than a real one. Additionally, we have three two-pan balances. It is known that two of the balances are accurate, while one is faulty (the result shown by the faulty balance has no connection to the actual weights of the coins placed on them and can be either correct or arbitrarily incorrect for different weighings). We do not know which balance is faulty. How can we identify the counterfeit coin in $3k + 1$ weighings?

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