
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 a 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 $f(x)$ on the interval $\left[\frac{1}{2}, 2\right]$.
(2) Suppose $a > 0$, and for any $x > 0$, $f(x) \geq f(1)$, compare the values of $\ln a$ and $-2b$.

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