
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 $F_{1}(-\sqrt{3},0)$, $F_{2}(\sqrt{3},0)$ are the foci of the ellipse $E: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, and point $P(1, \frac{\sqrt{3}}{2})$ is a point on the ellipse $E$.

$(1)$ Find the equation of the ellipse $E$;  
$(2)$ A line $l$ passing through point $F_{1}$ intersects the ellipse at points $M$ and $N$. Find the equation of line $l$ when the area of $\triangle F_{2}MN$ reaches its maximum value.

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