
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 $P(x, y)$ (where $x \neq 0$) is any point on the hyperbola defined by $\frac{y^2}{4} - x^2 = 1$, draw two perpendicular lines from point $P$ to each asymptote of the hyperbola. Let $A$ and $B$ represent the feet of these perpendicular lines. Determine the area of $\triangle PAB$.

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