
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 that the left and right foci of the hyperbola $C$ are $F_1$ and $F_2$, respectively, and $F_2$ is the focus of the parabola $y^{2}=8x$. Let $A$ be an intersection point of hyperbola $C$ and the parabola. If $\triangle AF_1F_2$ is an isosceles triangle with base $AF_1$, find the eccentricity of the hyperbola $C$.

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