
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 \( x^2 + z_1 x + z_2 + m = 0 \), where \( z_1, z_2, \) and \( m \) are complex numbers, it is given that \( z_1^2 - 4z_2 = 16 + 20i \). Suppose the two roots of this equation \( \alpha \) and \( \beta \) 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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