
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 $z=(m^{2}-3m)+(m^{2}-m-6)i$, for what values of the real number $m$ is the complex number $z$:  
$(1)$ a real number; $(2)$ a pure imaginary number; $(3)$ corresponding to a point located in the third quadrant of the complex plane.

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