
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 $z_{1}=m+i$ and $z_{2}=2+mi$, where $i$ is the imaginary unit and $m\in R$.  
$(1)$ If $z_{1}\cdot z_{2}$ is a pure imaginary number, find the value of $m$.  
$(2)$ If $z_{1}^{2}-2z_{1}+2=0$, find the imaginary part of $\frac{z_{2}}{z_{1}}$.

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