
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 point $M$ is the right vertex of the ellipse $C$: $3x^{2}+4y^{2}=12$, and points $A$ and $B$ are two different points on the ellipse $C$ (both different from point $M$) such that the product of the slopes of lines $MA$ and $MB$ is $\frac{1}{4}$.

(I) Find the eccentricity and the coordinates of the foci of ellipse $C$.
(II) Determine whether line $AB$ passes through a fixed point; if so, find the coordinates of the fixed point; if not, explain why.

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