
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
Determine the standard equations of the following hyperbolas:  
(1) The hyperbola which shares a common focus with the hyperbola $\frac{x^2}{16} - \frac{y^2}{4} = 1$ and passes through the point $(3\sqrt{2}, 2)$.  
(2) The hyperbola that has the foci of the ellipse $3x^2 + 13y^2 = 39$ and has the lines $y = \pm \frac{x}{2}$ as its asymptotes.

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