
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 ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a > b > 0$, the right focus is denoted by $F$, and the intersection point of the right directrix with the x-axis is denoted by $A$. There exists a point $P$ on the ellipse such that the perpendicular bisector of the segment $AP$ passes through $F$. Determine the range of values for the eccentricity of the ellipse.

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