
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 eccentricity $\frac{\sqrt{3}}{2}$, a line passing through the left focus $F$ and perpendicular to the $x$-axis intersects the ellipse $C$, resulting in a chord length of $1$. A line $l$ with slope $k (k \neq 0)$ passes through the point $(1,0)$ and intersects the ellipse $C$ at two distinct points $A$ and $B$.
- (I) Find the equation of the ellipse $C$;
- (II) Determine whether there exists a point $M$ on the $x$-axis such that $\overrightarrow{MA} \cdot \overrightarrow{MB} - \frac{k^2}{1+4k^2}$ is a constant value regardless of the value of $k$. If such a point exists, find the coordinates of $M$; if not, explain why.

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