
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 $z=a+bi\left(a,b\in R\right)$, where $i$ is the imaginary unit. If both $z+2i$ and $\frac{z}{{1-i}}$ are real numbers, find the complex number $z$. Furthermore, if the point corresponding to ${z_1}=\overline{z}+\frac{1}{{m-1}}-\frac{7}{{m+2}}i$ lies in the fourth quadrant, determine the range of real numbers for $m$.

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