SIMON-MARAIS 2023 A3

For each positive integer $n$, let $f(n)$ denote the smallest possible value of

$$|A_1 \cup A_2 \cup \cdots \cup A_n|,$$

where $A_1, A_2, \ldots, A_n$ 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 $n$.