
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 that one focus of the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1$ is $F(2,0)$, and the eccentricity is $\frac{\sqrt{6}}{3}$.
(I) Find the equation of the ellipse;
(II) A line passing through point $M(3,0)$ with slope $k$ intersects the ellipse at points $A$ and $B$. Point $C$ is the reflection of point $A$ over the $x$-axis. Find the maximum area of $\triangle MBC$.

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