
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 a positive integer \( n \), with \( k \) weights that each have integer weight and a balance scale, you can weigh items that have weights of \( 1, 2, 3, \ldots, n \) grams.

1. Determine the minimum value of \( k \), denoted as \( f(n) \).
2. Identify the specific values of \( n \) for which the composition of \( f(n) \) weights is uniquely determined. Provide a proof for your conclusion.

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