
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 $E$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\left(a > b > 0\right)$ with a vertex at $A(0,1)$ and a focal length of $2\sqrt{3}$.

$(\color{red}{Ⅰ})$ Find the equation of the ellipse $E$;

$(\color{red}{Ⅱ})$ A line passing through the point $P(-2,1)$ with a slope of $k$ intersects the ellipse $E$ at two distinct points $B$ and $C$. The lines $AB$ and $AC$ intersect the $x$-axis at points $M$ and $N$, respectively. Find the value of $k$ when $|MN|=2$.

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