
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 $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ with a focal length of $2$, and point $Q\left(\dfrac{a^2}{\sqrt{a^2 - b^2}}, 0\right)$ lies on the line $l$: $x = 3$.
$(1)$ Find the standard equation of ellipse $C$;
$(2)$ If $O$ is the origin, $P$ is a moving point on line $l$, and a line through point $P$ touches the ellipse at point $A$, find the minimum value of the area $S$ of $\triangle POA$.

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