
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 point $P$ and a non-zero real number $\lambda$, if two different lines $l_{1}$ and $l_{2}$ both pass through point $P$ and the product of their slopes is $\lambda$, then the lines $l_{1}$ and $l_{2}$ are called a "$P_{\lambda}$ conjugate pair" of lines. For example, the lines $l_{1}: y=2x$ and $l_{2}: y=-\frac{1}{2}x$ form an "$O_{-1}$ conjugate pair" of lines, where $O$ is the origin.<br/>$(1)$ Given that $l_{1}$ and $l_{2}$ form an "$O_{-3}$ conjugate pair" of lines, and $l_{1}: y=2x$, find the equation of line $l_{2}$;<br/>$(2)$ Given point $Q(-1,-\sqrt{2})$, lines $l_{1}$ and $l_{2}$ form a "$Q_{-2}$ conjugate pair" of lines. When the slope of $l_{1}$ varies, find the range of the product of the distances from the origin $O$ to lines $l_{1}$ and $l_{2}$.

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