
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 $m \in \mathbb{R}$, the complex number $z= \frac{m(m+2)}{m-1}+(m^2+2m-3)i$, for what value(s) of $m$ does:  
(1) $z$ belong to $\mathbb{R}$ (i.e., $z$ is a real number)?  
(2) $z$ belong to the set of imaginary numbers?  
(3) $z$ is a pure imaginary number?  
(4) The point corresponding to $z$ lie in the second quadrant of the complex plane?  
(5) The point corresponding to $z$ lie on the line $x+y+3=0$?

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