
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
Let $x=m$ and $x=n$ be the two extreme points of the function $f(X)=\ln x+ \frac{1}{2}{x}^{2}-(a+2)x$, where $m < n$, $a∈R$.

(1) If the tangent line of the curve $y=f(X)$ at point $P(1,f(1))$ is vertical to the $y$-axis, find the value of the real number $a$;

(2) Find the range of $f\left(m\right)+f\left(n\right)$;

(3) If $n\geqslant \sqrt{e}+ \frac{1}{ \sqrt{e}}-2$, find the maximum value of $f\left(n\right)-f\left(m\right)$ ($e$ is the base of the natural logarithm).

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