
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
Consider the set  $S=\{1,2,...,n\}$ . For every  $k\in S$ , define  $S_{k}=\{X \subseteq S, \ k \notin X, X\neq \emptyset\}$ . Determine the value of the sum \[S_{k}^{*}=\sum_{\{i_{1},i_{2},...,i_{r}\}\in S_{k}}\frac{1}{i_{1}\cdot i_{2}\cdot...\cdot i_{r}}\] <details><summary>Click to expand</summary>in fact, this problem was taken from an austrian-polish</details>

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