
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 $f(x) = x^2 - mx + 2\ln x$ where $m \in \mathbb{R}$.
(1) If $f(x)$ is monotonically increasing within its domain, find the range of real numbers for $m$;
(2) If $4 < m < 5$ and $f(x)$ has two extreme points $x_1$ and $x_2$, where $x_1 < x_2$, find the range of $f(x_1) - f(x_2)$.

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