
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 $f(x) = x\ln x - k(x-1)$, where $k \in \mathbb{R}$.  
$(1)$ When $k=1$, find the extreme value of the function $f(x)$.  
$(2)$ Is there a positive integer $k$ such that $f(x) + x > 0$ holds for all $x \in (1, +\infty)$? If yes, find the maximum value of $k$; if not, explain the reason.

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