
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 extremum of $-\frac{4}{3}$ at $x=1$, find $b$ and $c$;

$(2)$ Let the slope of the tangent line at any point $P$ on the graph of the function $y=f(x)-c(x+b)$, when $x \in (\frac{1}{2},3)$, be $k$. If $k \leqslant 2$, find the range of the real number $b$.

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