
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 $f(x)=\ln x+\frac{1}{2}{{x}^{2}}-ax+a(a∈R)$.

(1) If the function $f(x)$ is monotonically increasing on $(0,+∞)$, find the range of values for the real number $a$:

(2) If the function $f(x)$ takes extreme values at $x=x\_1$ and $x=x\_2$, where $x\_2 > x\_1$, and $a\geqslant {{e}^{\frac{1}{2}}}+{{e}^{-\\frac{1}{2}}}(e$ is the base of the natural logarithm$)$, find the maximum value of $f(x\_{2})-f(x\_{1})$.

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