
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 $f(x) = 2x - \frac{1}{x} - a\ln x$ where $a \in \mathbb{R}$,

(1) When $a = 3$, find the intervals of monotonicity for $f(x)$;
(2) Let $g(x) = f(x) - x + 2a\ln x$, and suppose $g(x)$ has two extremum points $x_1$, $x_2$. Determine the range of values for $a$.

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