
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 $z = (a^2 - 4) + (a + 2)i$ where $a \in \mathbb{R}$:  
(Ⅰ) If $z$ is a pure imaginary number, find the value of the real number $a$;  
(Ⅱ) If the point corresponding to $z$ in the complex plane lies on the line $x + 2y + 1 = 0$, find the value of the real number $a$.

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