
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
Let the complex number $z=(m^2-2m-3)+(m^2+3m+2)i$ where $m \in \mathbb{R}$. Determine the values of $m$ for which:

$(1)$ $z$ is a pure imaginary number.

$(2)$ The point corresponding to $z$ is located in the first quadrant of the complex plane.

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