
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 a complex number $z=(m^{2}+2m-8)+(m^{2}+2m-3)i$ where $m\in R$ corresponds to point $A$ in the complex plane.<br/>$(1)$ If the complex number $z-m+2$ is purely imaginary, find the value of $m$;<br/>$(2)$ If point $A$ is in the third quadrant, find the range of values for $m$.

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