
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 real number $m$ is the complex number $Z=\left( 2m^2-3m-2 \right)+\left( m^2-3m+2 \right)i$ 

$(1)$ a real number;  
$(2)$ an imaginary number;  
$(3)$ a pure imaginary number;  

$(4)$ the corresponding point is above the $x$-axis.

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