
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
The focal length of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ $(a>2, b>0)$ is $2c$ $(c>0)$. Given points $A(a,0)$, $B(0,b)$, and the point $(2,0)$ has a distance of $d_{1}$ to the line $AB$, and the point $(-2,0)$ has a distance of $d_{2}$ to the line $AB$, and $d_{1}+d_{2}\geqslant \frac{4}{5}c$, determine the range of eccentricity of the hyperbola.

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