
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 ellipse $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\left(a \gt b \gt 0\right)$ have left and right foci $F_{1}$, $F_{2}$, with the origin $O$ as the coordinate origin. The eccentricity of ellipse $C$ is $\frac{\sqrt{3}}{2}$.<br/>$(1)$ If the upper vertex of ellipse $C$ is $W$, and the area of $\triangle WF_{1}F_{2}$ is $9\sqrt{3}$, find the standard equation of ellipse $C$;<br/>$(2)$ Let a line $l$ with slope $k(0 \lt k\leqslant \frac{\sqrt{3}}{2})$ pass through an internal point $P(1,0)$ of ellipse $C$ and intersect $C$ at points $M$ and $N$. If there exists a point $Q$ on ellipse $C$ such that $\overrightarrow{OM}+\overrightarrow{ON}=\overrightarrow{OQ}$, find the maximum value of $b$.

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