
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 $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\left(a \gt b \gt 0\right)$, a point $A$ on the ellipse is symmetric with respect to the origin $O$ to point $B$. $F$ is the right focus of the ellipse. If $AF\bot BF$, $\angle ABF=\alpha$, and $α∈[{\frac{π}{12},\frac{π}{4}}]$, then the range of the eccentricity of the ellipse is ____.

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