
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 $x$.
1. If the function $f(x)$ has an extreme value of $-\frac{4}{3}$ at $x = 1$, find the values of $b$ and $c$.
2. If the slope $k$ of the tangent line at any point $P$ on the graph of the function $y = f(x) - c(x + b)$ satisfies $k \leq 2$ for $x \in (\frac{1}{2}, 3)$, find the range of values for the real number $b$.

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