
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
Ellipse $C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\left(a \gt b \gt 0\right)$, the line $y=k\left(x-1\right)$ intersects the ellipse $C$ at points $M$ and $N$ passing through one of the foci of the ellipse, and the point $A(1,\frac{3}{2})$ lies on the ellipse $C$. <br/>$(I)$ Find the equation of the ellipse $C$; <br/>$(II)$ If the perpendicular bisector of segment $MN$ intersects the $x$-axis at point $P$, is there a fixed point $Q$ on the $x$-axis such that $\frac{|PQ|}{|MN|}$ is a constant value? If so, find the coordinates of point $Q$ and the value of $\frac{|PQ|}{|MN|}$; if not, explain the reason.

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