
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
For the quadratic equation in $x$, \( x^2 + z_1 x + z_2 + m = 0 \), where \( z_1 \), \( z_2 \), and \( m \) are all complex numbers, and \( z_1^2 - 4z_2 = 16 + 20i \). Suppose the two roots of this equation are \( \alpha \) and \( \beta \), and they satisfy \( |\alpha - \beta| = 2 \sqrt{7} \). Determine the maximum and minimum values of \( |m| \).

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