
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
Find the standard equation of the hyperbola in the following cases:

(1) The hyperbola shares the same foci as the ellipse $\frac{x^{2}}{27} + \frac{y^{2}}{36} = 1$ and passes through the point $(\sqrt{15}, 4)$;

(2) The asymptotes of the hyperbola are given by the equations $2x \pm 3y = 0$, and the hyperbola passes through the point $P(\sqrt{6}, 2)$.

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