
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 $z = (a^2 + a - 2) + (a^2 - 7a + 6)i$, where $a \in \mathbb{R}$. For what value(s) of $a$ does:

(1) $z \in \mathbb{R}$?  
(2) $z$ is a pure imaginary number?  
(3) $z$ is zero?

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