
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 $C$: $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a > b > 0$) with its left focus at $(-\sqrt{3},0)$, and the ellipse $C$ passes through the point $(1,-\dfrac{\sqrt{3}}{2})$. Let line $l_{1}$ be $x-\sqrt{3}y+4=0$.

$(1)$ Find the standard equation of the ellipse $C$;

$(2)$ Suppose line $l$ is perpendicular to line $l_{1}$ and intersects the ellipse $C$ at points $P$ and $Q$. Find the maximum area of $\triangle OPQ$.

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