
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 the complex number $z=(m-1)(m+2)+(m-1)i$ where $m\in\mathbb{R}$ and $i$ is the imaginary unit.

1. If $z$ is a purely imaginary number, find the value of the real number $m$.
2. If $m=2$, let $\frac{z+i}{z-i}=a+bi$ where $a,b\in\mathbb{R}$, find the value of $a+b$.

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