
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 $f\left(x\right)=x^{3}+ax^{2}-6x+1\left(a\in R\right)$, and ${f'}\left(1\right)=-6$.  
$(1)$ Find the equation of the tangent line to the graph of the function $f\left(x\right)$ at the point $\left(1,f\left(1\right)\right)$;  
$(2)$ If the function $g\left(x\right)=f\left(x\right)-m$ has three zeros in the interval $\left[-2,4\right]$, find the range of real numbers for $m$.

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