
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: $a_6 = a_5 + 2a_4$, find the minimum value of $\frac{1}{m} + \frac{9}{n}$ if there exist two terms $a_m$ and $a_n$ such that $\sqrt{a_m a_n} = 2a_1$.

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