
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 the ellipse \( C \) with focus \( F \) and directrix \( l \), a line parallel to \( y = x \) passes through \( F \) and intersects ellipse \( C \) at points \( A \) and \( B \). It is known that the center \( P \) of the ellipse \( C \) lies within a circle for which \( AB \) is the diameter. Find the range of values for the eccentricity \( e \) of the ellipse \( C \).

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