
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 ellipse $C: \frac{y^2}{a^2} + \frac{x^2}{b^2} = 1 \left(a > b > 0\right)$ with eccentricity $\frac{\sqrt{3}}{2}$, and a circle centered at the origin with the radius equal to the semi-minor axis of the ellipse is tangent to the line $x-y+\sqrt{2}=0$. Let $A$ and $B$ be the right and top vertices of the ellipse $C$, respectively. The line $y=kx \left(k > 0\right)$ intersects the ellipse at points $E$ and $F$.

(Ⅰ) Find the equation of the ellipse $C$;

(Ⅱ) When the area of quadrilateral $AEBF$ is maximized, find the value of $k$.

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