
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 balance scales without weights and 3 coins that look identical, one of which is counterfeit: it is lighter than the real ones (the real coins are of the same weight). How many weighings are needed to identify the counterfeit coin? Solve the same problem in the cases where there are 4 coins and 9 coins.

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