
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 that all terms are positive in the geometric sequence $\{a_n\}$, and $a_2 \cdot a_4 = 4$, $a_1 + a_2 + a_3 = 14$, find the maximum positive integer value of $n$ such that $a_n \cdot a_{n+1} \cdot a_{n+2} > \frac{1}{9}$.

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