
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
Let the function $f(x)=\ln x+ \frac{1}{2}x^{2}-(m+2)x$, where $x=a$ and $x=b$ are two extreme points, with $a < b, m\in\mathbb{R}$.

$(1)$ Find the range of values for the real number $m$;

$(2)$ If $\frac{b}{a}\geqslant e$ (where $e$ is the base of the natural logarithm), find the maximum value of $f(b)-f(a)$.

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