
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 P with its center O at the origin of the coordinate system, its foci on the x-axis, and passing through point A(0, $2\sqrt{3}$), with an eccentricity of $\frac{1}{2}$:  
(1) Find the equation of ellipse P;  
(2) Determine whether there exists a line l passing through point E(0, -4) that intersects ellipse P at points R and T, satisfying $\overrightarrow{OR} \cdot \overrightarrow{OT} = \frac{16}{7}$. If such a line exists, find the equation of line l; if not, explain why.

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