
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 the hyperbola $E$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>0,b>0)$ with asymptotes $l_1:y=\frac{x}{2}$ and $l_2:y=-\frac{x}{2}$.  
$(1)$ Find the eccentricity of the hyperbola $E$.  
$(2)$ Let $O$ be the origin. A line $l$ passing through a point $P(2\sqrt{2},1)$ on the hyperbola intersects the lines $l_{1}$ and $l_{2}$ at points $A$ and $B$ respectively ($A$ and $B$ are in the first and fourth quadrants), and $\overrightarrow{PB}=2\overrightarrow{AP}$. Find the area of $\triangle AOB$.

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