
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 hyperbola \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\) (\(a > 1, b > 0\)) has a focal distance of \(2c\). Line \(l\) passes through the points \((a, 0)\) and \((0, b)\), and the sum of the distances from the points \((1,0)\) and \((-1,0)\) to line \(l\) is \(s \geqslant \frac{4}{5}c\). Determine the range of the eccentricity \(e\) of the hyperbola.

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