
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 the hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a>0, b>0)$ with a real axis length of $4\sqrt{2}$ and an eccentricity of $\frac{\sqrt{6}}{2}$. Point $P$ is any point on the hyperbola $C$.  
$(1)$ Find the standard equation of the hyperbola $C$;  
$(2)$ Given point $A(3,0)$, find the locus equation of the midpoint $Q$ of the line segment $AP$;  
$(3)$ Given point $A(3,0)$, find the minimum value of $|AP|$.

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