
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 that $i$ is the imaginary unit, $z_1=1-i$, and $z_2=3+ai$ ($a\in\mathbb{R}$).
(1) If $z_1+z_2$ is a real number, find the value of $z_1z_2$;
(2) If $\frac{z_2}{z_1}$ is a pure imaginary number, find $|z_2|$.

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