
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 the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ has an eccentricity of $\dfrac{1}{2}$, and a circle centered at the origin with the ellipse's minor axis as its radius is tangent to the line $x - y + \sqrt{6} = 0$. A line $l$ passing through point $P(4,0)$ and not perpendicular to the $x$-axis intersects the ellipse $C$ at points $A$ and $B$.

$(1)$ Find the equation of the ellipse $C$;

$(2)$ Find the range of values for $\overrightarrow{OA} \cdot \overrightarrow{OB}$.

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