
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 center of the ellipse $C$ is at the origin, its focus is on the $x$-axis, and its eccentricity is $\frac{1}{2}$. The ellipse passes through point $M\left(1, \frac{3}{2}\right)$.

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

(II) Determine if there exists a line $l_{1}$ passing through point $P(2,1)$ that intersects the ellipse $C$ at two distinct points $A$ and $B$ such that $\overrightarrow{PA} \cdot \overrightarrow{PB}={\overrightarrow{PM}}^{2}$. If it exists, find the equation of the line $l_{1}$; otherwise, explain the reason.

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