
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 $z$ is a complex number, and both $z+2i$ and $\frac{z}{{2-i}}$ are real numbers, where $i$ is the imaginary unit.<br/>$(1)$ Find the complex number $z$;<br/>$(2)$ If $z_{1}=\overline{z}+\frac{1}{m-1}-\frac{7}{m+2}i$ corresponds to a point in the fourth quadrant, find the range of real number $m$.

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