
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 $\{a_n\}$ be a geometric sequence, and $S_n$ be the sum of the first $n$ terms of $\{a_n\}$. Given that $S_3=14$, and $a_1+8$, $3a_2$, $a_3+6$ form an arithmetic sequence in that order, find the value of $a_1 \cdot a_3$.

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