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

(2) The length of the major axis of an ellipse is twice the length of its minor axis and it passes through point $A$ $(2,0)$. Find the standard equation of the ellipse.

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