
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 a hyperbola $C$: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ with an eccentricity of $e$, a line $x = \frac{a^{2}}{c}$ intersects with two asymptotes at points $P$ and $Q$, $F$ is the right focus, and $\triangle FPQ$ is an equilateral triangle.

(1) Find the eccentricity $e$ of the hyperbola $C$.
(2) If the chord length of the hyperbola $C$ intercepted by the line $y = ax + b$ is $\frac{b^{2}e^{2}}{a}$, find the equation of the hyperbola $C$.

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