
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 the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ with foci $F_1$ and $F_2$, and a point $P$ on the hyperbola, find the range of the eccentricity of the hyperbola, given that the angle bisector of $\angle F_1PF_2$ divides the line segment $F_1F_2$ in the ratio $5:1$.

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