
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 $f(x) = \begin{cases} x, & 0<x<1 \\ \frac{1}{x}, & x\geq 1 \end{cases}$, and $g(x) = af(x) - |x-1|$.  
(I) When $a=0$, if $g(x) \leq |x-2| + b$ holds for any $x \in (0, +\infty)$, find the range of values for the real number $b$;  
(II) When $a=1$, find the maximum value of $g(x)$.

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