
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 of the geometric sequence $\{a_n\}$ are positive and satisfy $a_7 = a_6 + 2a_5$, if there exist two different terms $a_m$ and $a_n$ such that $a_m \cdot a_n = 16a_1^2$, then the minimum value of $\frac{1}{m} + \frac{4}{n}$ is \_\_\_\_\_\_.

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