
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 an ellipse $E$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with a major axis length of $4\sqrt{3}$, and $P(2,\sqrt{2})$ is a point on $E$. <br/>$(1)$ Find the equation of $E$;<br/>$(2)$ If $A$ and $B$ are two points on $E$, and the coordinates of the midpoint of the line segment $AB$ are $(-\frac{8}{5},\frac{2}{5})$, find the value of $|AB|$.

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