
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 $C: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a>b>0)$ with an eccentricity $e= \frac{\sqrt{3}}{2}$, the distance between the origin and the line passing through points A($a$, $0$) and B($0$, $-b$) is $\frac{4\sqrt{5}}{5}$.

1. Find the equation of the ellipse $C$.
2. If the line $y=kx+1$ ($k \neq 0$) intersects the ellipse $C$ at two distinct points $E$ and $F$, and both $E$ and $F$ lie on the circle with center $B$, find the value of $k$.

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