
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 $z_1 = m^2 + 2m - 3 + (m - 1)i$ is a pure imaginary number $(m \in \mathbb{R})$.

$(1)$ Find the value of $m$;

$(2)$ If the complex number $z_2 = \frac{4 - 2i}{(1 + \frac{1}{4}z_1) \cdot i}$, find $|z_2|$.

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