
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 \( z_1, z_2, m \) are all complex numbers, and

\[ z_1^2 - 4z_2 = 16 + 20i \]

Let the roots of the equation be \(\alpha\) and \(\beta\), and \(|\alpha - \beta| = 2\sqrt{7}\). Find the maximum and minimum values of \(|m|\).

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