
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) The vertex of a parabola is at the origin, and its directrix is given by the equation $y=-1$. Find the standard equation of the parabola.

(2) A hyperbola shares the same foci with the ellipse $\frac{y^2}{36}+\frac{x^2}{27}=1$, and passes through the point $(\sqrt{15},4)$. Find the standard equation of the hyperbola.

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