
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
For each positive integer \( n \), let \( f(n) \) denote the smallest possible value of
\[
\left|A_{1} \cup A_{2} \cup \cdots \cup A_{n}\right|,
\]
where \( A_{1}, A_{2}, \ldots, A_{n} \) are sets such that \( A_{i} \nsubseteq A_{j} \) and \( \left|A_{i}\right| \neq\left|A_{j}\right| \) whenever \( i \neq j \).
Determine \( f(n) \) for each positive integer \( n \).

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