
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 ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ has an eccentricity of $\frac{\sqrt{3}}{2}$, and the point $(\sqrt{2}, \frac{\sqrt{2}}{2})$ lies on $C$.  
$(1)$ Find the equation of the ellipse $C$;  
$(2)$ If a line $l$ intersects the ellipse $C$ at points $P$ and $Q$, and $O$ is the origin with $OP \perp OQ$, find the minimum area of $\triangle OPQ$.

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