
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 $A_{1}$, $A_{2}$, $B_{1}$, $B_{2}$ are the endpoints of the real and imaginary axes of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}=1 (a > b > 0)$, and $P(4, \sqrt{2})$ is a point on the hyperbola that satisfies $k\_{A_{1}P} \cdot k\_{A_{2}P} = \frac{1}{4}$.
(1) Find the standard equation of the hyperbola;
(2) A line $l$ passing through the point $Q(2, 2)$ intersects the hyperbola at exactly one point. Find the equation of the line $l$.

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