
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 functions $f(x)=x\ln x,g(x)=\frac{ax^{2}}{2}$.
(1) Find the equation of the tangent line to the function $f(x)$ at $x=e$;
(2) If there exists at least one $x\_0\in[1,e]$ such that $f(x\_0) < g(x\_0)$, find the range of values for the real number $a$;
(3) Let $k\in\mathbb{Z}$ and $f(x) > (k-3)x-k+2$ holds true for all $x > 1$, find the maximum value of the integer $k$.

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