
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 $z_{1}=3-mi$ and $z_{2}=1+2i$, where $m\in\mathbb{R}$  
$(1)$ If $\frac{z_{1}}{z_{2}+i}$ is a pure imaginary number, find the value of the real number $m$;  
$(2)$ If $|z_{1}+z_{2}|=|z_{1}-z_{2}|$, find $\overline{z_{1}-z_{2}}$.

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