
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 point $M(1, \frac{3}{2})$. A line $l$ passing through point $P(2,1)$ intersects the ellipse $C$ at two distinct points $A$ and $B$.  
(Ⅰ) Find the equation of the ellipse $C$;  
(Ⅱ) Determine whether there exists a line $l$ such that $\overrightarrow{PA} \cdot \overrightarrow{PB} = \overrightarrow{PM}^2$. If it exists, find the equation of line $l$; if not, explain why.

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