
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 $f(x) = \left(a - \frac{1}{2}\right)x^2 + \ln x$, and $g(x) = f(x) - 2ax$ where $a \in \mathbb{R}$.
1. When $a = -\frac{1}{2}$, find the maximum and minimum values of $f(x)$ on the interval $\left[\frac{1}{e}, e\right]$.
2. If $g(x) < 0$ holds for all $x \in (2, +\infty)$, find the range of values for $a$.

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