
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 the positive geometric sequence $\{a\_n\}$, 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 $\frac{1}{m} + \frac{5}{n}$.

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