
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 numbers $z_{1}=a+2i$ and $z_{2}=1-ai$ ($a>0$, $i$ is the imaginary unit).  
$(1)$ If $a=1$, find $\frac{z_{1}}{z_{2} }$.  
$(2)$ If $z_{2}$ is a complex root of the real coefficient equation $x^{2}+mx+5=0$, find $|z_{1}z_{2}|$.

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