
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 the point $\left(1, \frac{3}{2}\right)$ is on the ellipse $C$: $\frac{{x}^{2}}{{a}^{2}}+ \frac{{y}^{2}}{{b}^{2}}=1\left(a > b > 0\right)$ and the eccentricity of the ellipse is $\frac{1}{2}$.
(1) Find the equation of the ellipse $C$;
(2) If $M$ is the right vertex of the ellipse $C$, and $A,B$ are two different points on the ellipse $C$ (both different from $M$ and satisfy that the product of the slopes of the lines $MA$ and $MB$ is $\frac{1}{4}$), determine whether the line $AB$ passes through a fixed point. If it does, find the coordinates of the fixed point. If not, explain the reason.

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