
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
A parabola has its vertex at the origin, and its directrix passes through one focus of the ellipse $C$: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ ($a > b > 0$) and is perpendicular to the major axis of the ellipse. It is known that one intersection point of the parabola and the ellipse is $(-\frac{2}{3}, \frac{2\sqrt{6}}{3})$.
(1) Find the equation of the parabola and the equation of the ellipse $C$;
(2) If a hyperbola shares the same foci with the ellipse $C$ and has $y=\pm\frac{4}{3}x$ as its asymptotes, find the equation of the hyperbola.

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