
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 $k$, does the complex number $z=({k}^{2}-3k-4)+({k}^{2}-5k-6)i$ satisfy the following conditions?

$(1)$ Real number;            $(2)$ Pure imaginary number;

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