
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
Given a hyperbola with the equation $\frac{x^2}{m} + \frac{y^2}{n} = 1$ and an eccentricity of $\frac{2\sqrt{3}}{3}$, and sharing a focus with the parabola $x^2 = 8y$, determine the equation of the hyperbola.

A) $\frac{y^2}{3} - x^2 = 1$
B) $\frac{x^2}{4} - \frac{y^2}{12} = 1$
C) $y^2 - \frac{x^2}{3} = 1$
D) $\frac{x^2}{12} - \frac{y^2}{4} = 1$

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