
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 that the right focus of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a＞0,b＞0)$ is $F$, the line $l$ passing through point $F$ with a slope of $k\left(k\neq 0\right)$ intersects the hyperbola at points $A$ and $B$. The perpendicular bisector of segment $AB$ intersects the $x$-axis at point $D$. If $|AB|≥\sqrt{3}|DF|$, then the range of eccentricity of the hyperbola is ____.

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