
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
A hyperbola is defined by the equation \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) where \(a > 1\) and \(b > 1\). The hyperbola has a focal distance of \(2c\). A 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 the line \(l\) is at least \(\frac{4c}{5}\). Determine the range of values for the eccentricity \(e\) of the hyperbola.

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