
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, where the lower vertex is $A$, the right vertex is $B$, the eccentricity $e = \frac{\sqrt{3}}{2}$, and the parabola $E: y = \frac{x^2}{8}$ with focus $F$. Let $P$ be a point on the parabola $E$, and the tangent line to $E$ at point $P$ is $l$, which is parallel to $AB$.  
$(1)$ Find the equation of line $l$;  
$(2)$ If $l$ intersects the ellipse $C$ at points $M$ and $N$, and the area of $\triangle FMN = \frac{5\sqrt{31}}{4}$, find the equation of $C$.

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