
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 an ellipse $\frac {x^{2}}{a^{2}} + \frac {y^{2}}{b^{2}} = 1$ ($a > b > 0$) with a point A on the ellipse and its symmetric point with respect to the origin O as point B. F is the right focus of the ellipse. If $AF \perp BF$, and let $\angle ABF = \alpha$, where $\alpha \in \left [ \frac {\pi}{12}, \frac {\pi}{4} \right ]$, then find the range of the eccentricity of the ellipse.

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