
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 any finite set \( S \), let \( f(S) \) be the sum of the elements of \( S \) (if \( S \) is empty then \( f(S) = 0 \) ). Find the sum over all subsets \( E \) of \( S \) of \(\frac{f(E)}{f(S)}\) for \( S = \{1, 2, \ldots, 1999\} \).

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