
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 $a \in \mathbb{R}$.
- (Ⅰ) When $a=1$, find the monotonic interval of $f(x)$.
- (Ⅱ) When the function $f(x)$ is decreasing on $[1,2]$, find the range of real numbers $a$.
- (Ⅲ) Let $g(x) = f(x) - x^{2}$. Does there exist a real number $a$ such that when $x \in (0,e]$ ($e$ is the base of the natural logarithm), the function $g(x)$ has a minimum value of $3$? 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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