
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
A geometric sequence with positive terms $\{a_n\}$ satisfies the relation $a_3 = a_2 + 2a_1$. If there exist terms $a_m$ 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}
user