
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 a complex number $z$ such that $z+2i\in R$, $\frac{z}{{2-i}}\in R$, where $i$ is the imaginary unit.<br/>$(1)$ Find the conjugate complex number $\overline{z}$ of $z$;<br/>$(2)$ If the point corresponding to the complex number $\left(z+mi\right)^{2}$ on the complex plane is in the fourth quadrant, find the range of real number $m$.

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