
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
Let ellipse $E$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ have an eccentricity of $\dfrac{1}{2}$. The minimum distance from a point $P$ on $E$ to the right focus is $1$.  
$(1)$ Find the equation of the ellipse $E$;  
$(2)$ A line passing through the point $(0,2)$ intersects the ellipse $E$ at two distinct points $A$ and $B$. Find the range of values for $\overrightarrow{OA} \cdot \overrightarrow{OB}$.

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