
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 $f(x) = x(\ln x + 1) + ax$;  
$(1)$ If $x=1$ is an extreme point of the function $f(x)$, find the value of $a$;  
$(2)$ If the function $f(x)$ is increasing on the interval $[\sqrt{e}, +∞)$, find the range of values for $a$;  
$(3)$ When $a=1$ and $k \in \mathbb{Z}$, the inequality $k(x - 1) < f(x) - x$ always holds for $x \in (1, +∞)$. Find the maximum value of $k$.

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