
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, $z-i$ is a real number, and $\frac{z-3i}{-2-i}$ is a pure imaginary number (where $i$ is the imaginary unit).  
$(1)$ Find the complex number $z$;  
$(2)$ If the point corresponding to the complex number $z_{1}=\frac{z}{m+i^{2023}}$ in the complex plane lies in the second quadrant, find the range of real number $m$.

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