
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 an eccentricity $e= \frac{\sqrt{6}}{3}$, and a latus rectum equation $x= \frac{3 \sqrt{6}}{2}$.
(1) Find the equation of the ellipse $C$;
(2) Let $G$ and $H$ be two moving points on the ellipse, and $O$ be the coordinate origin with $OG \perp OH$.
$\quad\quad\ ①$ When the slope angle of the line $OG$ is $60^{\circ}$, find the area of $\triangle GOH$;
$\quad\quad\ ②$ Does there exist a fixed circle with the origin $O$ as its center such that it is always tangent to the line $GH$? If so, find the equation of this circle; if not, explain the reason.

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