
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 line $l$ intersects with the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ at points $A$ and $B$, $M$ is the midpoint of segment $AB$, and the extension of $OM$ intersects with ellipse $C$ at point $P$.
(1) If the product of the slopes of line $l$ and line $OM$ is $-\frac{1}{4}$, and the major axis of the ellipse is $4$, find the equation of ellipse $C$;
(2) If quadrilateral $OAPB$ is a parallelogram, find the area of quadrilateral $OAPB$.

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