
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 $\dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1(a > b > 0)$, the sum of the distances from any point on the ellipse to the two foci $F_{1}$, $F_{2}$ is $4 \sqrt {2}$, and the eccentricity is $\dfrac { \sqrt {3}}{2}$.  
$(1)$ Find the standard equation of the ellipse;  
$(2)$ If the slope of line $l$ is $\dfrac {1}{2}$, and line $l$ intersects the ellipse $C$ at points $A$, $B$. Point $P(2,1)$ is a point on the ellipse, find the maximum value of the area of $\triangle PAB$.

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