
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 $f(x) = -\frac{1}{3}x^3 + bx^2 + cx + bc$ with respect to $x$, it is known that the function has an extreme value of $-\frac{4}{3}$ at $x=1$.
1. Find the values of $b$ and $c$.
2. Let $g(x) = ax^2 - 2\ln x (x > 0, a \in \mathbb{R})$. If there exist $x\_1$, $x\_2 \in (0, 3)$ such that $|f(x\_1) - g(x\_2)| < 1$, find the range of values for the real number $a$.

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