
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 $\dfrac{x^2}{a^2} - y^2 = 1\ (a > 0)$ with its right focus $F$, draw a line perpendicular to an asymptote of the hyperbola passing through point $F$. Let $P$ be the foot of the perpendicular. If the $y$-coordinate of point $P$ is $\dfrac{2\sqrt{5}}{5}$, find the eccentricity of this hyperbola.

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