
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
In the rectangular coordinate system $(xOy)$, there is an ellipse $(C)$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with an eccentricity $e = \frac{\sqrt{2}}{2}$, and the length of the minor axis is $2\sqrt{3}$.

(1) Find the standard equation of the ellipse $(C)$.

(2) If point $P(2, 1)$ lies on the ellipse $(C)$, and moving points $A$ and $B$ both lie on the ellipse $(C)$, with the midpoint $M$ of $AB$ on line segment $OP$ (excluding endpoints). Find the maximum area of triangle $AOB$.

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