
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 $E$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\left(a \gt b \gt 0\right)$ with an eccentricity of $\frac{\sqrt{2}}{2}$ and a vertex $A(0,1)$. <br/>$(Ⅰ)$ Find the equation of the ellipse $E$; <br/>$(Ⅱ)$ If the line $l$ passing through point $A$ intersects the ellipse $E$ at another point $B$ such that $|AB|=\frac{4}{3}\sqrt{2}$, find the coordinates of point $B$.

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