
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 the minimum value of the function $f(x)$ is 3 when $x \in (0, e]$ ($e$ is the base of the natural logarithm); if such a $b$ exists, find its value; if not, explain why.

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