
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 \(3 x^{2} - y^{2} + 24 x + 36 = 0\) has its right focus at \(F\) and its right directrix at \(l\). An ellipse \(C\) takes \(F\) and \(l\) as its corresponding focus and directrix. A line parallel to \(y = x\) passes through \(F\) and intersects the ellipse \(C\) at points \(A\) and \(B\). Given that the center of \(C\) lies within a circle whose diameter is \(AB\), determine the range of values for the eccentricity of the ellipse \(C\).

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