
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
Let the hyperbola $C$ be defined by $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>0,b>0)$. Points $A_{1}$ and $A_{2}$ are the left and right vertices of the hyperbola $C$, respectively, and point $P$ lies on the hyperbola $C$.  
$(Ⅰ)$ If $A_{1}A_{2}=4b$ and point $P(2\sqrt{2},-1)$, find the equation of the hyperbola $C$.  
$(Ⅱ)$ When $P$ is different from points $A_{1}$ and $A_{2}$, and the product of the slopes of lines $PA_{1}$ and $PA_{2}$ is $2$, find the eccentricity of the hyperbola $C$.

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