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

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