
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
In a geometric sequence $\left\{a_{n}\right\}$ with all positive terms, there exist two terms $a_{m}$ and $a_{n}$ such that $\sqrt{a_{m} a_{n}}=8 a_{1}$, and it is known that $a_{9}=a_{8}+2 a_{7}$. Find the minimum value of $\frac{1}{m}+\frac{4}{n}$.

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