
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
$(1)$ Given a complex number $z$ corresponding to a point in the second quadrant of the complex plane, with $|z|=2$ and $z+\overline{z}=-2$, find $z$;<br/>$(2)$ Given a complex number $z=\frac{{2{m^2}}}{{1-i}}-({1+2i})m-3({2+i})$ is purely imaginary, find the real value of $m$.

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