
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
Let the function $f(x) = x^2 - x\ln x - 2$.
(Ⅰ) Determine the intervals of monotonicity for $f(x)$.
(Ⅱ) If there exists an interval $[a,b] \subseteq \left[\frac{1}{2}, +\infty\right)$ such that the range of $f(x)$ on $[a,b]$ is $[k(a+2), k(b+2)]$, find the range of values for $k$.

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