
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 center of the hyperbola $C$ is at the origin, its foci are on the coordinate axes, and $P(1, -2)$ is a point on $C$, and $y= \sqrt {2}x$ is one of the asymptotes of $C$, derive the equation of $C$.

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