
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 $f(x)=x\ln x+ax$ ($a\in\mathbb{R}$).
- $(1)$ If the function $f(x)$ is decreasing on the interval $[e,e^{2}]$, find the range of values for $a$.
- $(2)$ If for any $x\in(1,+\infty)$, $f(x) > k(x-1)+ax-x$ always holds, find the maximum value of the positive integer $k$.

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