
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 $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\left(a>b>0\right)$ with an eccentricity of $\frac{1}{2}$, a line passing through the left focus intersects the ellipse at points $P$ and $Q$, with $|PQ|=3$.

(I) Find the equation of ellipse $C$;

(II) Let $M$ be the right vertex of ellipse $C$, and let $A$ and $B$ be two distinct points on the ellipse (both different from $M$) such that the product of the slopes of lines $MA$ and $MB$ is $\frac{1}{4}$. Determine whether line $AB$ passes through a fixed point. If it does, find the coordinates of that point; if not, explain why.

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