
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 the complex number $z=(a^2-4)+(a+2)i$ where $a \in \mathbb{R}$.

(I) If $z$ is a purely imaginary number, find the value of the real number $a$.

(II) If the point corresponding to $z$ on the complex plane is 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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