
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 $z_1=3-mi$, $z_2=1+2i$, 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