
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 $f(x)=ax-\ln x$.  
$(1)$ Find the monotonic interval of the function $f(x)$.  
$(2)$ If there exists $a \gt 0$ such that $f(x)≥\frac{{{a^2}}}{2}+b$ holds for any $x\in \left(0,+\infty \right)$, find the range of real number $b$.

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