
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 $f(x)=\frac{1}{2}x^2+bx+\ln x$,

(1) If the function $f(x)$ is monotonically increasing in its domain, find the range of values for the real number $b$.

(2) Let $g(x)=f(x)-bx-\frac{1+a}{2}x^2$, where $a\in R$, discuss the monotonic intervals of $g(x)$.

(3) If under the condition of (1), $f(x)\leq x^2+\frac{1}{2x^2}-3x+1$ holds for all $x\in (0,1]$, find the range of values for the real number $b$.

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