
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
An ellipse $E$: $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt b \gt 0\right)$ is known to have a vertex $A\left(0,-2\right)$. The area of the quadrilateral formed by the four vertices of the ellipse $E$ is $4\sqrt{5}$.

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

$(\color{red}{Ⅱ})$ A line passing through the point $P\left(0,-3\right)$ with slope $k$ intersects the ellipse $E$ at two distinct points $B$ and $C$. The lines $AB$ and $AC$ intersect the line $y=-3$ at points $M$ and $N$, respectively. Find the range of values for $k$ such that $|PM|+|PN|\leqslant 15$.

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