
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
There exists $x_0 \in (2, +\infty)$, such that $k(x_0 - 2) > x_0 (\ln x_0 + 1)$, then the minimum value of the positive integer $k$ is \_\_\_\_\_\_.
(Reference data: $\ln 2 \approx 0.6931$, $\ln 3 \approx 1.0986$, $\ln 5 \approx 1.6094$)

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