
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
Let points $M$ and $N$ be two different points on the ellipse $C$: $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt b \gt 0\right)$, and the length of segment $MN$ is at most $4$. The eccentricity of the ellipse is $e=\frac{\sqrt{3}}{2}$.<br/>$(1)$ Find the standard equation of the ellipse $C$;<br/>$(2)$ If the line $MN$ passes through the point $Q(0,2)$ and $\overrightarrow{OM}•\overrightarrow{ON} \gt 0$, and the midpoint of segment $MN$ is $P$, find the range of slopes of the line $OP$.

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