
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 an ellipse $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $e = \dfrac{1}{2}$, and passing through the point $(\sqrt{3}, \dfrac{\sqrt{3}}{2})$

$(I)$ Find the equation of ellipse $C$;  
$(II)$ Through point $A(-a,0)$, there are two perpendicular lines $l_1$ and $l_2$ intersecting ellipse $C$ at another point $P$ and $Q$, respectively. Does the line $PQ$ pass through a fixed point? If so, find this fixed point; otherwise, explain why.

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