
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 $f(x)=\frac{1}{3}{x}^{3}+a{x}^{2}+6x$, when $x=3$, the function $y=f\left(x\right)$ reaches an extremum.  
$(1)$ If $f\left(x\right)$ is increasing on $\left(m,m+2\right)$, find the range of real numbers for $m$;  
$(2)$ If $1\leqslant x\leqslant 3$, the equation $f\left(x\right)+m=0$ has two roots, find the range of real numbers for $m$.

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