
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 triangle $\triangle ABC$, given vectors $\overrightarrow{m}=(\sin A, \cos A)$, $\overrightarrow{n}=(1, -\sqrt{3})$, and $\overrightarrow{m} \cdot \overrightarrow{n}=0$.  
$(1)$ Find the value of $\tan A$;  
$(2)$ If $\triangle ABC$ is an acute triangle and satisfies $\frac{m}{\tan A}=\frac{1}{\tan B}+\frac{1}{\tan C}$, find the minimum value of the real number $m$.

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