
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
In the Cartesian coordinate system $xOy$, it is known that the ellipse $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ has a focal length of $2$.
$(1)$ If the ellipse $C$ passes through the point $\left(\dfrac{\sqrt{6}}{2}, 1\right)$, find the standard equation of the ellipse $C$;
$(2)$ Let $A(-2,0)$, and $F$ be the left focus of the ellipse $C$. If there exists a point $P$ on the ellipse $C$ such that $\dfrac{PA}{PF} = \sqrt{2}$, find the range of the eccentricity of the ellipse $C$.

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