
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 plane $xOy$, it is known that ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \, (a > b > 0)$ passes through point $P(2, 1)$, and its eccentricity $e = \frac{\sqrt{3}}{2}$.

$(1)$ Find the equation of ellipse $C$;
$(2)$ If line $l$ is given by the equation $y = \frac{1}{2}x + m$, and this line intersects with ellipse $C$ at points $A$ and $B$, find the maximum value of the area of triangle $PAB$.

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