
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 the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ ($a > 0$, $b > 0$) with left and right foci $F_1$ and $F_2$ respectively. On the hyperbola's asymptote, there exists a point $P(3, 4)$ such that $PF_1 \perp PF_2$. Determine the hyperbola's equation.

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

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