
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 $f(x)=ax+x\ln x$ $(a\in \mathbb{R})$,

$(1)$ If the function is increasing in the interval $[e,+\infty)$, find the range of values for $a$;

$(2)$ If $a=1$, when $k\in \mathbb{Z}$, the inequality $k(x-1) < f(x)$ always holds for $x\in (1,+\infty)$. Find the maximum value of $k$.

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