
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 an eccentricity $e= \frac{\sqrt{3}}{2}$, and a point $P(0, \frac{3}{2})$ whose distance to the right focus $(F)$ of the ellipse $(C)$ is $\frac{\sqrt{57}}{2}$. Let line $(l)$ pass through point $P$ with an existing slope, intersecting the ellipse $(C)$ at points $A$ and $B$. The perpendicular bisector of segment $AB$ intersects the $x$-axis at point $Q$.

(I) Find the standard equation of the ellipse $(C)$;
(II) Find the range of values for the $x$-coordinate $x_0$ of point $Q$.

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