
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 an arithmetic sequence $\{a_n\}$ with positive terms and a common ratio $q=2$, if there exist two terms $a_m$ and $a_n$ such that $\sqrt{a_m \cdot a_n} = 4a_1$, find the minimum value of $\frac{1}{m} + \frac{4}{n}$.

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