
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
Let  $F$  be the set of all  $n$ -tuples  $(A_1, \ldots, A_n)$  such that each  $A_{i}$  is a subset of  $\{1, 2, \ldots, 1998\}$ .  Let  $|A|$  denote the number of elements of the set  $A$ .  Find
\[ \sum_{(A_1, \ldots, A_n)\in F} |A_1\cup A_2\cup \cdots \cup A_n|  \]

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