
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
For  $a>0$ , denote by  $S(a)$  the area of the part bounded by the parabolas  $y=\frac 12x^2-3a$  and  $y=-\frac 12x^2+2ax-a^3-a^2$ .
Find the maximum area of  $S(a)$ .

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