
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 $f\left(x\right)=\ln{x}+\frac{1}{2}ax^{2}+bx,a,b∈R$.

(1) If $a=-2$, $b=1$, find the monotonic intervals of the function $f\left(x\right)$;

(2) If for any $a∈[1,+∞)$, $f\left(x\right)≥0$ always holds in the domain $x∈[1,+∞)$, find the range of values for the real number $b$.

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