
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 $f(x) = \ln x - x$.
$(1)$ Find the equation of the tangent line to the curve $y = f(x)$ at $x = e$.
$(2)$ Find the intervals of monotonicity and extreme values of the function $g(x) = f(x) + 2x - 4\ln x - \frac{2}{x}$.
$(3)$ If the inequality $f(x) \leq (a-1)x + 1$ holds for all $x \in (0, +\infty)$, find the range of real numbers for $a$.

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