
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  $\dfrac{f(E)}{f(S)}$  for  $S=\{1,2,\cdots,1999\}$ .

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