
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 $e=\frac{\sqrt{2}}{2}$. Also, point $P(2,1)$ is on the ellipse $(C)$.

1. Find the equation of the ellipse $(C)$.
2. If points $A$ and $B$ are both on the ellipse $(C)$, and the midpoint $M$ of $AB$ is on the line segment $OP$ (not including the endpoints), find the maximum value of the area of triangle $AOB$.

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