
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 $f(x) = x - \frac{2a-1}{x} - 2a\ln x$ ($a \in \mathbb{R}$),  
(Ⅰ) If the function $f(x)$ has an extremum at $x=2$, find the value of the real number $a$;  
(Ⅱ) If $f(x) \geq 0$ holds for any $x \in [1, +\infty)$, find the range of values for the real number $a$.

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