
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
Let the ellipse $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1 (a > b > 0)$ have its left focus at $F$, eccentricity $\dfrac{\sqrt{3}}{3}$, and the line segment intercepted by the ellipse on the line passing through point $F$ and perpendicular to the $x$-axis is $\dfrac{4\sqrt{3}}{3}$.  
(I) Find the equation of the ellipse;  
(II) Let $A$ and $B$ respectively be the left and right vertices of the ellipse. A line passing through point $F$ with slope $k$ intersects the ellipse at points $C$ and $D$. If $\overrightarrow{AC} \cdot \overrightarrow{DB} + \overrightarrow{AD} \cdot \overrightarrow{CB} = 7$, find the value of $k$.

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