
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
Given a complex number $z = (a^{2} - 7a + 12) + (a^{2} - 5a + 6)i$ where $a \in \mathbb{R}$, for what value(s) of $a$ is $z$ a real number? For what value(s) of $a$ is $z$ an imaginary number? For what value(s) of $a$ is $z$ a pure imaginary number?

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