
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 $C:\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1(a > 0,b > 0)$, a circle with center at the right focus of $C$, $F(c,0)$, and radius $a$ intersects with one asymptote of $C$ at points $A$ and $B$. If $|AB|=\frac{2}{3}c$, find the eccentricity of the hyperbola $C$.

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