
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 $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 1)$ with a focal length of $2$, and the area of the circle passing through one endpoint of the minor axis and the two foci is $\dfrac{4}{3}\pi$. A line $l$ with slope $k (k \neq 0)$ passing through the right focus of the ellipse $C$ intersects the ellipse at points $A$ and $B$, and the midpoint of the line segment $AB$ is $P$.
$(1)$ Find the standard equation of the ellipse $C$;
$(2)$ The line passing through point $P$ and perpendicular to $AB$ intersects the $x$-axis at point $D$, and $|DP| = \dfrac{3\sqrt{2}}{7}$, find the value of $k$.

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