
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
Let $A_{n}=\left\{\frac{1}{2},\frac{3}{4},\frac{5}{8},\ldots,\frac{2n-1}{2^{n}}\right\}(n\in\mathbb{N}^{*},n\geqslant 2)$, and let $S$ be the sum of the smallest elements of all non-empty subsets of $A_{n}$. Find the value of $S$.

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