
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 $f(x)=\ln (x-2)-\frac{x^{2}}{2a}$ (where $a$ is a constant, $a\neq 0$).
(I) When $a=1$, find the equation of the tangent line to the function $f(x)$ at the point $(3,f(3))$.
(II) Find the monotonic intervals of $f(x)$.
(III) If $f(x)$ takes an extreme value at $x_{0}$, and $x_{0} \notin [e+2,e^{3}+2]$, while $f(x)\geqslant 0$ always holds on $[e+2,e^{3}+2]$, find the range of values for the real number $a$. (Here, $e$ is the base of the natural logarithm.)

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