
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
In the equation \( x^{2} + z_{1} x + z_{2} + m = 0 \) with respect to \( x \), where \( z_{1}, z_{2}, m \) are all complex numbers, and \( z_{1}^{2} - 4z_{2} = 16 + 20i \), suppose the two roots of this equation are \( \alpha \) and \( \beta \), which satisfy \( |\alpha - \beta| = 2 \sqrt{7} \). Find the maximum and minimum values of \( |m| \).

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