
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)\\). 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}
user