
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]


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Given the two foci of an ellipse $F_{1}(-\sqrt{3}, 0)$, $F_{2}(\sqrt{3}, 0)$, and a line $l_{1}$ passing through $F_{1}$ and not parallel to the coordinate axes intersects the ellipse at points M and N. If the perimeter of $\triangle MNF_{2}$ equals 8.  
(I) Find the equation of the ellipse;  
(II) If a line $l$ passing through the point $(1, 0)$ intersects the ellipse at two distinct points P and Q, is there a fixed point E $(m, 0)$ on the x-axis such that $\overrightarrow{PE} \cdot \overrightarrow{QE}$ is always a constant? If it exists, find the coordinates of E and the constant value; if not, explain why.

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