
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 $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1$ ($a > 0, b > 0$) with its right focus at $F(c,0)$, and the circle $F$: $(x-c)^{2} + y^{2} = c^{2}$, the line $l$ is perpendicular to one of the asymptotes of the hyperbola $C$ and has an $x$-axis intercept of $\dfrac{2}{3}a$. If the chord length of the circle $F$ cut by the line $l$ is $\dfrac{4\sqrt{2}}{3}c$, calculate the eccentricity of the hyperbola.

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