
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 that the complex number $z$ is purely imaginary, and $\frac{z+2}{1-i}+z$ is a real number, where $i$ is the imaginary unit.<br/>$(1)$ Find the complex number $z$;<br/>$(2)$ If the point represented by the complex number $\left(m-z\right)^{2}$ is in the first quadrant, find the range of real number $m$.

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