
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
In the set of real numbers $\mathbb{R}$, the differentiable function $f(x) = \frac{1}{3}x^3 + \frac{1}{2}ax^2 + 2bx + c$ has a maximum value point $x\_1 \in (0, 1)$ and a minimum value point $x\_2 \in (1, 2)$, determine the range of values for $\frac{b-2}{a-1}$.

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