
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 point F is the right focus of the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 (a > 0, b > 0)$, and the line $x = a$ intersects with the asymptote of the hyperbola in the first quadrant at point A. If the midpoint of AF lies on the hyperbola, calculate the eccentricity of the hyperbola.

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