
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 $f(x) = \ln x - ax (a \in \mathbb{R})$.

$(1)$ If for any $x$ in the domain of $f(x)$, we have $f(x) \leqslant 0$, find the range of values for the real number $a$;

$(2)$ If $a=1$, let the function $g(x) = f(x) + \frac{1}{2}x^2 - bx$. Suppose $x_1$ and $x_2 (x_1 < x_2)$ are the two extreme points of the function $g(x)$. If $b \geqslant \frac{3}{2}$ and $g(x_1) - g(x_2) \geqslant k$ always holds, find the maximum value of the real number $k$.

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