
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 numbers $z\_1=a^2-3+(a+5)i$ and $z\_2=a-1+(a^2+2a-1)i$, where $a \in \mathbb{R}$, which correspond to the vectors $\overrightarrow{OZ\_1}$ and $\overrightarrow{OZ\_2}$ respectively (with $O$ being the origin). If the vector $\overrightarrow{Z\_1 Z\_2}$ corresponds to a purely imaginary number, find the value of $a$.

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