
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 with the equation $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1 (a > b > 0)$ and an eccentricity of $\frac{\sqrt{3}}{2}$. Also, point $P(2,1)$ lies on the ellipse.

(1) Find the standard equation of the ellipse.

(2) If the slope of line $l$ is $\frac{1}{2}$, and line $l$ intersects the ellipse $C$ at points $A$ and $B$, find the maximum area of triangle $PAB$.

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