
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 a complex number $z=m+2+(m-2)i$ where $m\in \mathbb{R}$, $\overline{z}$ is the conjugate of $z$, and $z+\overline{z}=6$.  
$(1)$ Find the value of $m$;  
$(2)$ If $z-3i$ is a root of a real-coefficient quadratic equation in $x$, $x^{2}+ax+b=0$, find the other complex root of this quadratic equation.

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