
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 with \( k \) weights each weighing a positive integer number of grams and a balance scale, it is possible to measure all weights from \( 1 \) to \( n \) grams.
(1) Find the minimum value of \( k \), denoted as \( f(n) \).
(2) For which values of \( n \) is the composition of the \( f(n) \) weights unique? Provide a proof for your answer.

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