
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 $z_{1}=1+i$ and $z_{2}=2+mi$ where $m\in \mathbb{R}$.  
$(1)$ If $\frac{{z}_{2}}{{z}_{1}}$ is purely imaginary, find $m$.  
$(2)$ If $\frac{{z}_{2}}{{z}_{1}}\in \mathbb{R}$, find the sum of the real part and the imaginary part of $3z_{1}+iz_{2}$.

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