
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 $f\left(x\right)=ax^{3}+bx^{2}+cx+2$ has an extremum at $x=2$, its graph passes through the point $\left(2,-6\right)$, and ${f'}\left(0\right)=-4$.  
$(1)$ Find the analytical expression of the function $f\left(x\right)$;  
$(2)$ Find the equation of the tangent line of the function $f\left(x\right)$ at $x=-1$.

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