
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 a complex number $z=m^{2}+m-2+\left(m-1\right)i\left(m\in R\right)$, where $i$ is the imaginary unit.<br/>$(1)$ If $z$ is a purely imaginary number, find the value of the real number $m$;<br/>$(2)$ If $m=2$, and $\frac{z+i}{\overline{z}-i}=a+bi\left(a,b\in R\right)$, find the value of $a+b$.

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