
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
The first term of the geometric sequence $\{a_n\}$ is $2$, and the common ratio is $3$. The sum of the first $n$ terms is $S_n$. If $\log_{3} \left( \frac{1}{2}a_n(S_{4m}+1)\right)=9$, then the minimum value of $\frac{1}{n}+ \frac{4}{m}$ is ______.

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