
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 each positive integer  $ n$ , define  $ f(n)$  as the number of digits  $ 0$  in its decimal representation. For example,  $ f(2)\equal{}0$ ,  $ f(2009)\equal{}2$ , etc. Please, calculate \[ S\equal{}\sum_{k\equal{}1}^{n}2^{f(k)},\] for  $ n\equal{}9,999,999,999$ .
*Yudi Satria, Jakarta*

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