
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 $f\left( x \right)=x\ln x-\frac{a}{2}{{x}^{2}}$, the line $l$: $y=\left( k-2 \right)x-k+1$, and $k\in Z$.

(1) If there exists ${{x}_{0}}\in \left[ e,{{e}^{2}} \right]$ such that $f\left( {{x}_{0}} \right) > 0$ holds, find the range of values for the real number $a$.

(2) If $a=0$, find the maximum value of $k$ such that the graph of the function $f\left( x \right)$ is always above the line $l$ when $x > 1$.

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