
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 point $A$ and $F(c,0)$ be the right vertex and right focus of the hyperbola $\frac{x^{2}}{a^{2}}- \frac{y^{2}}{b^{2}}=1$ ($a > 0$, $b > 0$). The line $x= \frac{a^{2}}{c}$ intersects one of the asymptotes of this hyperbola at point $P$. Given that $\triangle PAF$ is an isosceles triangle, determine the eccentricity of this hyperbola.

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