
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 $E$ with the equation $\frac{x^2}{a^2}+ \frac{y^2}{b^2}=1$ $(a > b > 0)$ and eccentricity $\frac{\sqrt{2}}{2}$, and a circle $C$ with the equation $(x-2)^2+(y-1)^2=\frac{20}{3}$. If $E$ and $C$ intersect at points $A$ and $B$, and the line segment $AB$ is exactly the diameter of circle $C$.
(1) Find the equation of the line $AB$;
(2) Find the standard equation of the ellipse $E$.

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