
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 with eccentricity $\dfrac{\sqrt{6}}{3}$ and equation $\dfrac{x^{2}}{a^{2}}+ \dfrac{y^{2}}{b^{2}}=1$ $(a > b > 0)$, one of its foci is $F$. A line passing through $F$ and perpendicular to the $x$-axis intersects the ellipse at points $A$ and $B$, with $|AB| = \dfrac{2\sqrt{3}}{3}$.  

$(1)$ Find the equation of this ellipse;

$(2)$ Given a line $y=kx+2$ intersects the ellipse at points $C$ and $D$. If a circle with diameter $CD$ passes through point $E(-1,0)$, find the value of $k$.

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