
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 $f(x) = a\ln x + \frac{1-a}{2}x^2 - x$ ($a \in \mathbb{R}$ and $a \neq 1$), if there exists $x_0 \in [1, +\infty)$ such that $f(x_0) < \frac{a}{a-1}$, determine the range of values for $a$.

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