
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 quadratic equation \(x^{2} + z_{1} x + z_{2} + m = 0\), the coefficients \(z_{1}\), \(z_{2}\), and \(m\) are complex numbers, and it is given that \(z_{1}^{2} - 4 z_{2} = 16 + 20i\). Suppose the two roots \(\alpha\) and \(\beta\) of this equation 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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