
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ passes through the point $A(-\frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2})$ and has an eccentricity of $\frac{\sqrt{2}}{2}$. Let $F_1$ and $F_2$ be its left and right foci, respectively.  
$(1)$ Find the standard equation of the ellipse $C$;  
$(2)$ Suppose there exist two points $M$ and $N$ on the parabola $y^2 = 4x$, and two points $P$ and $Q$ on the ellipse such that $M$, $N$, $F_2$ are collinear, and $P$, $Q$, $F_2$ are collinear, and also $PQ \perp MN$. Find the minimum value of the area of quadrilateral $PMQN$.

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