
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 Cartesian coordinate system $(xOy)$, an ellipse $(C)$ is defined by the equation $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity of $e = \frac{\sqrt{2}}{2}$. The point $P(2, 1)$ lies on the ellipse $(C)$.

(1) Find the equation of the ellipse $(C)$;
(2) If points $A$ and $B$ both lie on the ellipse $(C)$ and the midpoint $M$ of $AB$ lies on the line segment $OP$ (excluding endpoints).
$\quad\quad$ (a) Find the slope of the line $AB$;
$\quad\quad$ (b) Find the maximum area of $\triangle AOB$.

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