
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 eccentricity $\frac{{\sqrt{6}}}{3}$, the point $({\frac{{2\sqrt{2}}}{3},\frac{{\sqrt{3}}}{3}})$ lies on the ellipse $C$. <br/>$(1)$ Find the standard equation of the ellipse $C$; <br/>$(2)$ A line passing through the point $N\left(2,0\right)$ intersects the ellipse $C$ at points $A$ and $B$. Find the maximum value of $S_{\triangle AOB}$.

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