
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
(1) Given that a hyperbola shares the same foci with the ellipse $\frac{x^2}{27} + \frac{y^2}{36} = 1$ and intersects the ellipse at a point with the y-coordinate of 4, find the equation of the hyperbola.  
(2) Find the equation of the hyperbola that shares the same asymptotes with the hyperbola $\frac{x^2}{4} - \frac{y^2}{3} = 1$ and passes through the point (2, 3).

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