
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 $z=m^2(1+i) - m(3+i) - 6i$, determine for which real number $m$ the complex number $z$ is:  
(1) A real number; (2) A complex number; (3) A pure imaginary number; (4) Zero; (5) Corresponding to a point in the third quadrant.

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