
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) = x^2^ + 2ax and g(x) = 4a^2^lnx + b, suppose the curves y = f(x) and y = g(x) have a common point P, and their tangents at point P are the same. When a ∈ (0, +∞), the maximum value of the real number b is _____.

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