
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 that one focus of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ coincides with the focus $F$ of the parabola $y^{2}=4 \sqrt{3}x$, and the two endpoints of the minor axis of the ellipse form an equilateral triangle with $F$.
(1) Find the equation of the ellipse;
(2) If the line $l$ passing through the point $(1,0)$ intersects the ellipse at two distinct points $P$ and $Q$, does there exist a fixed point $E(m,0)$ on the $x$-axis such that $\overrightarrow{PE} \cdot \overrightarrow{QE}$ is always constant? If so, find the coordinates of $E$ and the constant value; if not, explain the reason.

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