
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
If the foci of two hyperbolas are on the same coordinate axis and the ratio of their conjugate axis to their transverse axis is equal, then they are called parallel hyperbolas. Given that hyperbola $M$ is parallel to the hyperbola $\frac{x^2}{16} - \frac{y^2}{4} = 1$, and the point $(2,0)$ is on hyperbola $M$.

(I) Find the equation of hyperbola $M$;

(II) Let point $P$ be any point on hyperbola $M$, and point $A$ has coordinates $(3,0)$. Find the minimum value of $|PA|$.

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