
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
For the hyperbola \(\frac{x^{2}}{4} - \frac{y^{2}}{12} = 1\), the right focus is \(F\) and the left directrix is \(l\). The ellipse \(C\) takes \(F\) as one focus and \(l\) as its directrix. A line with a slope of 1 passes through \(F\) and intersects the ellipse \(C\) at points \(A\) and \(B\). If the center \(P\) of the ellipse \(C\) lies within a circle with diameter \(A B\), determine the range of values for the eccentricity \(e\) of the ellipse \(C\).

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