
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 $f(x)=ax-1-\ln x$ where $a > 0$,
$(1)$ Find the interval of monotonicity for the function $f(x)$;
$(2)$ If the function $f(x)$ has an extremum at $x=1$, and for any $x \in (0, +\infty)$, the inequality $f(x) \geqslant bx-2$ always holds, find the maximum value of the real number $b$.

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