
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 $z=(m^{2}+5m+6)+(m^{2}-2m-15)i$ where $m\in\mathbb{R}$, find the value(s) of $m$ such that:
$(1)$ $z$ is a real number?
$(2)$ The point corresponding to $z$ lies in the third quadrant?

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