
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 $k \in \mathbb{R}$, and the function $f(x) = \ln x - kx$.

$(1)$ If $k=2$, find the equation of the tangent line to the curve $y=f(x)$ at point $P(1, -2)$.

$(2)$ If the equation $f(x) = 0$ has no roots, find the range of values for the real number $k$.

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