
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 one focus of the hyperbola $S$ is the same as the focus of the parabola $y^2 = 12x$, and if $y = -\sqrt{2}x$ is an asymptote of the hyperbola $S$, then the equation of the hyperbola $S$ is __________.

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