
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 with two foci $F_{1}(-\sqrt{3},0)$, $F_{2}(\sqrt{3},0)$, and the two endpoints of the minor axis of the ellipse form an equilateral triangle with $F_{2}$.  
$(I)$ Find the equation of the ellipse;  
$(II)$ A line $l$ passing through the point $(1,0)$ and not parallel to the coordinate axes intersects the ellipse at two distinct points $P$ and $Q$. If there exists a fixed point $E(m,0)$ on the x-axis such that $\overrightarrow{PE} \cdot \overrightarrow{QE}$ is always a constant, find the value of $m$.

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