
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 $f(x)=-x^{3}+ax^{2}+bx+c$ has a tangent line at point $P(1,-2)$ with the equation $y=-3x+1$.

$(1)$ If the function $f(x)$ has an extremum at $x=-2$, find the expression of $f(x)$;

$(2)$ If the function $f(x)$ is monotonically increasing in the interval $[-2,0]$, find the range of the real number $b$.

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