
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
For what value of the real number $m$ is the complex number $z= \frac {m^{2}+m-6}{m}+(m^{2}-2m)i$  
$(1)$ a real number?  
$(2)$ an imaginary number?  
$(3)$ a pure imaginary number?

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