
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 $\{a_n\}$ where each term is positive, there exist two terms $a_{m}$ and $a_{n}$ $(m, n \in \mathbb{N}^{*})$ such that $\sqrt{a_{m}a_{n}} = 4a_{1}$, and $a_{7} = a_{6} + 2a_{5}$, determine the minimum value of $\frac{1}{m} + \frac{4}{n}$.

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