
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 $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$, its eccentricity is $\frac{\sqrt{6}}{2}$, and the distance from its left vertex to one of its asymptotes is $\frac{2\sqrt{6}}{3}$. Determine the standard equation of this hyperbola.

A) $\frac{x^{2}}{12} - \frac{y^{2}}{8} = 1$
B) $\frac{x^{2}}{16} - \frac{y^{2}}{8} = 1$
C) $\frac{x^{2}}{16} - \frac{y^{2}}{12} = 1$
D) $\frac{x^{2}}{8} - \frac{y^{2}}{4} = 1$

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