
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
(1) Given $$\frac{\overline{z}}{1+i} = 2+i$$, find $z$.
(2) Given $m \in \mathbb{R}$, the complex number $z = \frac{m(m+2)}{m-1} + (m^2+2m-3)i$, for what values of $m$ is $z$ purely imaginary?

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