
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
In the complex plane, let $z = m^{2} - m - 6 + (m^{2} - 2m - 3)i$, where $m \in \mathbb{R}$.  
$(1)$ If the complex number $z$ is purely imaginary, find the value of $m$.  
$(2)$ If the point corresponding to the complex number $z$ is in the second quadrant, find the range of real numbers for $m$.

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