
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 $Z=\lg (m^{2}-2m-2)+(m^{2}+3m+2)i$. Find the values of $m$ for which:  
$(1) Z$ is a real number;  
$(2) Z$ is a pure imaginary number;  
$(3) 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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