
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 $f(x) = \ln x - mx, m \in \mathbb{R}$.

$(1)$ Find the intervals of monotonicity for $f(x)$;

$(2)$ If $f(x) \leqslant \frac{m-1}{x} - 2m + 1$ holds true over the interval $[1, +\infty)$, find the range of values for the positive real number $m$.

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