
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 a geometric sequence with positive terms $\{a_n\}$ where the first term $a_1=2$, and the sum of the first $n$ terms is $S_n$, it satisfies $2a_3+S_2=4$. Find the maximum positive integer value of $n$ such that $\frac{66}{65} < \frac{S_{2n}}{S_{n}} < \frac{16}{15}$.

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