
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
$(1)$ Among the three conditions: ① $z+\overline{z}=-2$, ② $z$ is purely imaginary, ③ $z$ is a non-zero real number, choose one and supplement it in the following problem, then solve the problem.<br/>Given the complex number $z=(m^{2}-4m+3)+(m^{2}-3m+2)i$, where $i$ is the imaginary unit, and $\overline{z}$ is the conjugate of $z$. If ____. Find the value of the real number $m$.<br/>($Note: If multiple conditions are chosen and answered separately, marks will be given according to the first condition chosen.)<br/>$(2)$ If $\omega =3-4i$ is a root of the real-coefficient one-variable quadratic equation: $x^{2}+ax+b=0$ with respect to $x$, find the values of $a$ and $b$, as well as the other root of the equation.

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