
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 $l_{1}$ and $l_{2}$ are two perpendicular lines passing through the point $P(-\sqrt{2}, 0)$, and each of $l_{1}$ and $l_{2}$ intersects the hyperbola $y^{2}-x^{2}=1$ at two points, respectively $A_{1}, B_{1}$ and $A_{2}, B_{2}$.
(1) Find the range of values for the slope $k_{1}$ of $l_{1}$.
(2) If point $A_{1}$ is exactly one of the vertices of the hyperbola, find the value of $\left|A_{2} B_{2}\right|$.
(3) If $\left|A_{1} B_{1}\right|=\sqrt{5}\left|A_{2} B_{2}\right|$, find the equations of $l_{1}$ and $l_{2}$.

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