
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 $C$ with its center at the origin and its foci on the $x$-axis, the eccentricity of $C$ is $\frac{1}{2}$, and it passes through the point $M(1, \frac{3}{2})$.

$(1)$ Find the equation of the ellipse $C$;

$(2)$ Is there a line $l_{1}$ passing through point $P(2,1)$ that intersects the ellipse $C$ at two distinct points $A$ and $B$, satisfying $\overrightarrow{PA} \cdot \overrightarrow{PB} = \overrightarrow{PM}^{2}$? If it exists, find the equation of the line $l_{1}$; if not, explain why.

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