
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 $f(x)=x^{2}+ax-\ln x$, where $a\in R$.
(I) Find the monotonic intervals of $f(x)$ when $a=1$;
(II) Find the range of values of the real number $a$ when the function $f(x)$ is a decreasing function on $[1,2]$;
(III) Let $g(x)=f(x)-x^{2}$. Determine if there exists a real number $a$ such that the minimum value of the function $g(x)$ is $3$ when $x\in(0,e]$ ($e$ is the base of the natural logarithm). If it exists, find the value of $a$; if not, explain the reason.

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