
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 $C$: $\frac{y^{2}}{a^{2}}+\frac{x^{2}}{b^{2}}=1 (a > b > 0)$ with center $O$, a circle with center $O$ and radius $\sqrt{\frac{ab}{2}}$ is called the "accompanying" circle of the ellipse. It is known that the eccentricity of the ellipse is $\frac{\sqrt{3}}{2}$, and it passes through the point $(\frac{1}{2}, \sqrt{3})$.

1. Find the equations of the ellipse $C$ and its "accompanying" circle.
2. A line $l$ is drawn through point $P(0, m)$ tangent to the "accompanying" circle, intersecting the ellipse $C$ at points $A$ and $B$. Denote the area of $\triangle AOB$ ($O$ is the coordinate origin) as $S_{\triangle AOB}$. Express $S_{\triangle AOB}$ as a function of $m$ and find its maximum value.

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