
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 $E$: $\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1(a > 0,b > 0)$ with its right vertex $A$, and the parabola $C$: $y^{2}=8ax$ with its focus $F$. If there exists a point $P$ on the asymptote of $E$ such that $\overrightarrow{AP}\bot \overrightarrow{FP}$, determine the range of the eccentricity of $E$.

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