
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
Ellipse $C$: $\dfrac{x^2}{4} + y^2 = 1$, $A(\sqrt{3}, \dfrac{1}{2})$, $B(-\sqrt{3}, -\dfrac{1}{2})$, find the product of the slopes of lines $PA$ and $PB$ for a point $P$ on ellipse $C$.

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