
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 $z\_1 = 1 - 3i$ and $z\_2 = a - 3i$ where $a \in \mathbb{R}$:

1. If $a = 2$, find $z\_1 \cdot \overline{z\_2}$.
2. If $z = \frac{z\_1}{z\_2}$ is a purely imaginary number, find the value of $a$.

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