
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
Given \( n \) silver coins labeled from 1 to \( n \). One of these coins may be counterfeit, differing in weight from the genuine coins, either lighter or heavier. The genuine coins have identical weight. Additionally, a standard silver coin labeled 0 is provided. How can you determine which coin is counterfeit and whether it is lighter or heavier using a balance scale the least number of times, where the number of times refers to the worst-case scenario or the average number of times the scale is used?

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