
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
Let $F$ be the focus of the parabola $C_1$: $y^2=2px$ ($p > 0$), and point $A$ be a common point of the parabola and an asymptote of the hyperbola $C_2$: $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2}=1$ ($a > 0, b > 0$). If line $AF$ is perpendicular to the x-axis, determine the eccentricity of the hyperbola.

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