
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
In the geometric sequence $\{a_n\}$, $(a_1 \cdot a_2 \cdot a_3 = 27)$, $(a_2 + a_4 = 30)$, $(q > 0)$

Find:
$(1)$ $a_1$ and the common ratio $q$;

$(2)$ The sum of the first $6$ terms $(S_6)$.

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