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

  $(1)$ 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 line $l_{1}$; if not, explain why.

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