APMO 1998 - Asian Pacific Mathematics Olympiad Question 1

Let $F$ be the set of all $n$-tuples $(A_1,A_2,\ldots,A_n)$ in which every $A_i$ is a subset of $\{1,2,\ldots,1998\}$. Write $|A|$ for the cardinality of a set $A$. Find
$$\sum_{(A_1,\ldots,A_n)\in F}|A_1\cup A_2\cup\cdots\cup A_n|.$$