
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 function $f(x) = \ln(2ax+1) + \frac{x^{3}}{3} - x^{2} - 2ax \ (a \in \mathbb{R})$.
1. If $y=f(x)$ is an increasing function on the interval $[3, +\infty)$, determine the range of values for the real number $a$.
2. When $a=-\frac{1}{2}$, the function $y=f(1-x) - \frac{(1-x)^{3}}{3} - \frac{b}{x}$ has a zero, find the maximum value of the real number $b$.

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