
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 of $\frac{\sqrt{6}}{3}$, and the distance from one endpoint of the minor axis to the right focus is $\sqrt{3}$.
(1) Find the equation of the ellipse $(C)$;
(2) The line $y=x+1$ intersects the ellipse at points $A$ and $B$. $P$ is a point on the ellipse. Find the maximum area of $\triangle PAB$.

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