
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
(1) Find the standard equation of the hyperbola with a focus on the $x$-axis, $c=\sqrt{6}$, and passing through the point $(-5,2)$.
(2) Given two points $P\_1$, $P\_2$ on the hyperbola with coordinates $(3,-4\sqrt{2}), (\frac{9}{4}, 5)$, find the standard equation of the hyperbola.

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