
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) with an eccentricity of $\frac{\sqrt{6}}{3}$, the area of the triangle formed by one endpoint of the minor axis and the two foci of the ellipse is $\frac{5\sqrt{2}}{3}$.  
(I) Find the equation of ellipse $C$;  
(II) A moving line $y = k(x + 1)$ intersects the ellipse $C$ at points $A$ and $B$. If the x-coordinate of the midpoint of segment $AB$ is $-\frac{1}{2}$, find the value of the slope $k$.

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