
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 $Z = \log(m^2 - 2m - 2) + (m^2 + 3m + 2)i$. Determine 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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