
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 $z=m-i$ where $m\in \mathbb{R}$, and $\overline{z}•(1+3i)$ is an imaginary number ($\overline{z}$ is the conjugate of $z$).<br/>$(1)$ Find the value of the real number $m$;<br/>$(2)$ Let $z_{1}=\frac{m+4i}{1-i}$, find $|z_{1}|$;<br/>$(3)$ Let $z_{2}=\frac{a-{i}^{2023}}{z}$, and the point corresponding to $z_{2}$ in the complex plane is in the first quadrant, find the range of real number $a$.

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