
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 C with an eccentricity of $\frac{5}{3}$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a>0$, $b>0$), where $F_1$ and $F_2$ are the left and right foci, respectively. Point $P$ is an intersection point of the directrix of the parabola $y^2 = 12x$ and the asymptote of hyperbola C. Additionally, $PF_1 \perp PF_2$. Determine the equation of the hyperbola.

A) $\frac{x^2}{16} - \frac{y^2}{9} = 1$
B) $\frac{x^2}{3} - \frac{y^2}{4} = 1$
C) $\boxed{\frac{x^2}{9} - \frac{y^2}{16} = 1}$
D) $\frac{x^2}{4} - \frac{y^2}{3} = 1$

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