
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
Given the sum of the first $n$ terms $S_{n}$ of a geometric sequence $\{a_{n}\}$, where $a_{n} \gt 0$, if ${a}_{1}{a}_{3}=16$ and ${S}_{1},\frac{3}{4}{S}_{2},\frac{1}{2}{S}_{3}$ form an arithmetic sequence, calculate $S_{6}$.

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