
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 $f(x) = ax + \ln(x-1)$, where $a$ is a constant.  
(1) If $h(x) = f(x+1)$, discuss the interval where $h(x)$ is monotonic.  
(2) If $a = \frac{1}{1-e}$ and there exists $x$ such that the inequality $\sqrt{f^2(x)} - \frac{e}{e-1} \leq \frac{2\ln x + bx}{2x}$ holds, determine the range of real numbers for $b$.

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