
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 complex number $z_{1}=m-2i$, complex number $z_{2}=1-ni$, where $i$ is the imaginary unit, and $m$, $n$ are real numbers.  
$(1)$ If $n=1$ and $z_{1}$ is a purely imaginary number, find the value of $|z_{1}+z_{2}|$;  
$(2)$ If $z_{1}=(\overline{{z}_{2}})^{2}$, find the values of $m$ and $n$.

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