
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
Consider a positive geometric sequence $\{a\_n\}$ that satisfies $a\_3 = a\_2 + 2a\_1$. If there exist $a\_m$ and $a\_n$ such that $a\_m \cdot a\_n = 64a\_1^2$, find the minimum value of $\frac{1}{m} + \frac{9}{n}$.

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