
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 that the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{4} = 1$ ($a > 0$) has an eccentricity of $\frac{\sqrt{5}}{2}$, and the points $F_1$ and $F_2$ are its foci on the left and right side, respectively. Point $P$ has coordinates $(5, y_0)$ and point $Q$ is the point on the hyperbola symmetrical to $P$ with respect to the origin. Find the area of the quadrilateral $F_1QF_2P$.

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