
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 $z_{1}=\frac{2{m}^{2}}{1-i}$, $z_{2}=\left(2+i\right)m-3\left(1+2i\right)$, $m\in R$, $i$ is the imaginary unit.<br/>$(Ⅰ)$ If $z_{1}+z_{2}$ is purely imaginary, find the value of the real number $m$;<br/>$(Ⅱ)$ If $z_{1}+z_{2} \gt 0$, find the value of $z_{1}\cdot z_{2}$.

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