
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 $z = (1+i)m^2 + (5-2i)m + (6-15i)$;  
(1) For what value(s) of the real number $m$ is $z$ a real number?  
(2) For what value(s) of the real number $m$ is $z$ a pure imaginary number?  
(3) For what value(s) of the real number $m$ does the corresponding point of $z$ lie on the line $x+y+7=0$?

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