
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
S is the set of all possible n-tuples \((X_1, X_2, \ldots, X_n)\) where each \(X_i\) is a subset of \{1, 2, \ldots, 1998\}. For each member \(k\) of S, let \(f(k)\) be the number of elements in the union of its \(n\) elements. Find the sum of \(f(k)\) over all \(k\) in S.

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