
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 asymptotic lines of a hyperbola are $y = \pm 2x$ and it passes through the point $(-3, 4\sqrt{2})$.
(1) Find the equation of the hyperbola;
(2) If the line $4x - y - 6 = 0$ intersects the hyperbola at points $A$ and $B$, find the value of $|AB|$.

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