
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 $(a^2 - 1) + (a + 1)i$, where $a \in \mathbb{R}$ and $i$ is the imaginary unit, is a pure imaginary number.

(1) Find the value of the real number $a$;

(2) If the complex number $z = \frac{a + \sqrt{3}i}{ai}$, find $|z|$.

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