
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 parabola $y^{2}=8x$ whose directrix intersects the $x$-axis at point $A$, and its focus is $F$. $P$ is any point on the parabola. When the ratio $\dfrac{|PF|}{|PA|}$ reaches its minimum value, point $P$ is exactly on the hyperbola with foci $A$ and $F$. Determine the eccentricity of this hyperbola.

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