
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 $a=-1$, $b=3$, find the maximum and minimum values of the function $f(x)$ on the interval $\left[\frac{1}{2},2\right]$.
- $(2)$ When $a=0$, determine whether there exists a positive real number $b$ such that, when $x\in(0,e]$ ($e$ is the base of the natural logarithm), the minimum value of the function $f(x)$ is $3$. If it exists, find the value of $b$; if not, explain why.

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