
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 $g(x)=3a^{2}\ln x+b$, defined on the set of positive real numbers, where $a > 0$. The curves $y=f(x)$ and $y=g(x)$ have a common point, and their tangents at this point are identical.

(1) If $a=1$, find the value of $b$.
(2) Express $b$ in terms of $a$ and find the maximum value of $b$.

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