
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 the right focus of the hyperbola $\dfrac {x^{2}}{a^{2}} - \dfrac {y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ be $F$, and its right vertex be $A$. A perpendicular line to $AF$ passing through $F$ intersects the hyperbola at points $B$ and $C$. Perpendicular lines to $AB$ and $AC$ passing through $B$ and $C$, respectively, intersect at point $D$. If the distance from $D$ to line $BC$ is not less than $a+c$, determine the range of the eccentricity of the hyperbola.

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