
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   $$ |A_1 \cup A_2 \cup \dots \cup A_n| $$   where  $A_1, A_2, A_3 \dots A_n$  are sets such that  $A_i \not\subseteq A_j$  and  $|A_i| \neq |A_j|$  whenever  $i \neq j$ . Determine  $f(n)$  for each positive integer  $n$ .

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