
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  $n{}$  coins, one of which is fake, which differs in weight from the real ones and a two-pan scale which works correctly if the weights on the pans are different, but can show any outcome if the weights on the pans are equal. For what  $n{}$  can we determine which coin is fake and whether it is lighter or heavier than the real coins, in at most  $k{}$  weightings?

*Proposed by A. Zaslavsky*

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