
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
The points $A_1$, $A_2$, $B_1$, $B_2$ are respectively the endpoints of the real and imaginary axes of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > b > 0$). Point $P(4, \sqrt{2})$ lies on the hyperbola, and it satisfies $k_{A_1P} \cdot k_{A_2P} = \frac{1}{4}$.  
(1) Find the standard equation of the hyperbola;  
(2) The line $l$ passing through point $Q(2, 2)$ intersects the hyperbola at exactly one point. Find the equation of line $l$.

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