
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 with positive terms $\{a_n\}$ that satisfies $a_7=a_6+2a_5$. If there exist two terms $a_m$, $a_n$ such that $\sqrt{a_ma_n}=4a_1$, calculate the minimum value of $\frac{1}{m}+ \frac{9}{n}$.

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