
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 functions $f(x) = \ln x$, $g(x) = \frac{1}{2}x^2 - 2x$, for $x > 2$, if $k(x - 2) < xf(x) + 2g'(x) + 3$ always holds, the maximum integer value of $k$ is ____.

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