
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 $C$: $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1 (a > b > 0)$, where its left vertex and right focus are $A$ and $F$, respectively, and its right directrix is $m$.

$(1)$ If there is no point $Q$ on line $m$ such that $\triangle AFQ$ is an isosceles triangle, find the range of the eccentricity of the ellipse.

$(2)$ Under the condition of $(1)$, when $e$ takes the maximum value, the coordinate of point $A$ is $(-2,0)$. Let $B$, $M$, and $N$ be three points on the ellipse, and $\overrightarrow{OB} = \dfrac{3}{5}\overrightarrow{OM} + \dfrac{4}{5}\overrightarrow{ON}$. Find the equation of the circle with the midpoint of line segment $MN$ as its center and passing through points $A$ and $F$.

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