
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 $f(x) = x\ln x - ax^2 + a$ ($a \in \mathbb{R}$), its derivative is $f'(x)$.

$(1)$ Find the extreme value of the function $g(x) = f'(x) + (2a-1)x$;

$(2)$ When $x > 1$, the inequality $f(x) < 0$ always holds. Find the range of values for $a$.

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