
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 all real numbers $x$, let $\lfloor x \rfloor$ be the greatest integer less than or equal to $x$. Then, the function $f(x)=\lfloor x \rfloor$ is called the Gauss function or the floor function. If $a_{n}=f\left( \frac{n}{2} \right), n \in \mathbb{N}^{*}$, then the sum of the first $2n$ terms of the sequence $\{2^{a_{n}}\}$, denoted as $S_{2n}=$ ______.

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