
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
Selected Exercise $(4-5)$: Inequality Lecture

Given the function $f(x)=\begin{cases} x,&0 < x < 1 \\\ \frac{1}{x},&x\geqslant 1 \\\ \end{cases}$, and $g(x)=af(x)-|x-1|$.

$(1)$ When $a=0$, find the range of real numbers $b$ such that $g(x)\leqslant |x-2|+b$ holds true for all $x\in (0,+\infty )$.

$(2)$ When $a=1$, find the maximum value of $g(x)$.

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