
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
For what value(s) of the real number $k$ is the complex number $z = (k^2 - 3k - 4) + (k^2 - 5k - 6)i$ a: (1) real number; (2) imaginary number; (3) purely imaginary number; (4) zero?

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