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