
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
Let $F_1$ and $F_2$ be the left and right foci of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, respectively. If there exists a point $P$ on the line $x= \frac{a^2}{c}$ (where $c^2+b^2=a^2$) such that the perpendicular bisector of segment $PF_1$ passes through point $F_2$, determine the range of the eccentricity of the ellipse.

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