
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 eccentricity $\frac{\sqrt{3}}{2}$, where the left and right foci are $F_{1}$ and $F_{2}$ respectively, and $P$ is the upper vertex of $C$. Also, $\triangle PF_{1}F_{2}$ has a perimeter of $4+2\sqrt{3}$.<br/>$(Ⅰ)$ Find the equation of the ellipse $C$;<br/>$(Ⅱ)$ Let a line $l$ passing through the fixed point $M\left(0,2\right)$ intersect the ellipse $C$ at two distinct points $A$ and $B$, and $\angle AOB$ is acute (where $O$ is the origin). Find the range of slope $k$ of the line $l$.

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