
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 $m\in \mathbb{R}$.  
$(1)$ When $m=1$, let ${z_1}=\frac{{z+2i}}{{1-i}}$, find the value of $|z_{1}|$;  
$(2)$ If $\overline{z}(1+3i)$ is an imaginary number ($\overline{z}$ is the conjugate of $z$), find the value of the real number $m$.

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