
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 foci on the \( x \)-axis, has an eccentricity of \(\frac{1}{2}\), and passes through the point \( M(1, \frac{3}{2}) \). A line \( l \) passing through the point \( P(2,1) \) intersects the ellipse \( C \) at two distinct points \( A \) and \( B \).
(1) Find the equation of the ellipse \( C \);
(2) Determine if there exists a line \( l \) such that \(\overrightarrow{P A} \cdot \overrightarrow{P B} = \overrightarrow{P M}^{2}\). If such a line exists, find the equation of the line \( l \); if it does not exist, explain why.

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