
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) Given that a hyperbola passes through point $A(6,-5)$, and one of its foci is at $(-6,0)$, find the standard equation of the hyperbola.

(2) Given that hyperbola $E$ shares the same asymptotes with the hyperbola: $\frac{x^{2}}{4}-\frac{y^{2}}{2}=1$, and the distance from the right focus $F(c,0)$ of hyperbola $E$ to the asymptote is $2$, find the standard equation of hyperbola $E$.

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