
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 points $F_1$ and $F_2$ which are the left and right foci of the hyperbola $\frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1 \quad (a > 0, b > 0),$ a line passing through $F_1$ and perpendicular to the x-axis intersects the hyperbola at points $A$ and $B$. If $\triangle ABF_2$ is an acute-angled triangle, determine the range of the eccentricity $e$ of the hyperbola.

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