
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 a hyperbola with the equation $$\frac {x^{2}}{a^{2}} - \frac {y^{2}}{b^{2}} =1$$ when a line with a slope of 2 passes through its right focus, it intersects both branches of the hyperbola at one point each; when the slope of the line is 3, it intersects the right branch of the hyperbola at two distinct points. Find the range of the hyperbola's eccentricity.

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