
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 the functions $f(x)= \frac{1}{2}x^{2}+2ax$ and $g(x)=3a^{2}\ln x+b$, assume that the curves $y=f(x)$ and $y=g(x)$ have a common point, and that the tangent lines at that point are the same, find the maximum value of the real number $b$ when $a\in(0,+\infty)$.

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