
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 $f(x)=ax^{2}+bx-\ln x (a,b \in \mathbb{R})$.

(1) When $a=-1$ and $b=3$, find the maximum and minimum values of the function $f(x)$ in the interval $\left[\frac{1}{2}, 2\right]$.

(2) When $a=0$, does there exist a positive real number $b$ such that the minimum value of the function $f(x)$ is $3$ when $x \in (0, e]$ ($e$ is the base of the natural logarithm)? If so, find the value of $b$; if not, explain the reason.

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