
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 $f(x) = \ln x + bx - c$, the equation of the tangent line to $f(x)$ at the point $(1, f(1))$ is $x + y + 4 = 0$.

$(1)$ Find the expression for $f(x)$;

$(2)$ Determine the intervals of monotonicity for $f(x)$;

$(3)$ If in the interval $\left[\frac{1}{2}, 3\right]$, it always holds that $f(x) \geqslant 2\ln x + kx$, find the range of values for $k$.

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