
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 $C: \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ (where $a > 0$ and $b > 0$), with its left and right foci being $F_1$ and $F_2$, respectively. Suppose that there exists a point $P$ on the hyperbola $C$ such that the triangle $PF_1F_2$ is isosceles with $\cos\angle PF_1F_2 = \frac{1}{8}$. Find the eccentricity of the hyperbola.

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