
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
Answer the following two questions:  
$(1)$ Hyperbola $E$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>0,b>0)$ has an eccentricity of $\sqrt{2}$, and the point $(2,\sqrt{2})$ lies on the hyperbola $E$. Find the equation of $E$.  
$(2)$ The real axis length of hyperbola $C$ is $2$, and hyperbola $C$ shares the same foci with the ellipse $\frac{x^2}{8}+\frac{y^2}{4}=1$. Find the standard equation of hyperbola $C$.

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