
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 $f(x) = \frac{1}{2}ax^2 - 2ax + \ln(x)$ where $a \neq 0$.
1. Discuss the monotonicity of $f(x)$.
2. If there exists an $x_0 \in \left[1+ \frac{\sqrt{2}}{2}, 2\right]$ such that the inequality $f(x_0) + \ln(a+1) > b(a^2 - 1) - (a+1) + 2\ln(2)$ holds true for any $1 < a < 2$, find the range of the real number $b$.

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