
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 asymptotes of the hyperbola, whose foci lie on the x-axis, pass through the intersection points of the ellipses \(\frac{x^{2}}{4}+\frac{y^{2}}{16}=1\) and \(\frac{a x^{2}}{16}+\frac{y^{2}}{4}=1\) (where \(0 < a \leq 1\)), find the range of the eccentricity of the hyperbola.

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