
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 $\frac{x^{2}}{16+k} - \frac{y^{2}}{8-k} = 1 \ (-16 < k < 8)$ with one asymptote equation $y = -\sqrt{3}x$, and points $P(3, y_{0})$ and $Q$ are symmetric about the origin on the hyperbola, calculate the area of the quadrilateral $F_{1}QF_{2}P$.

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