
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 \( n \), it is known that using \( k \) weights, each with a positive integer weight in grams, and a balance scale, it is possible to measure objects with masses of \( 1, 2, 3, \cdots, n \) grams.
(1) Find the minimum value of \( k \), denoted as \( f(n) \).
(2) For which values of \( n \) is the composition of these \( f(n) \) weights uniquely determined? Provide a proof of your conclusion.

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