
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
Given the quadratic equation in $x$ defined by $x^{2} + z_{1} x + z_{2} + m = 0$, where $z_{1}, z_{2}, m$ are complex numbers and $z_{1}^{2} - 4 z_{2} = 16 + 20i$, assume the roots of this equation are $\alpha$ and $\beta$, and they satisfy $|\alpha - \beta| = 2\sqrt{7}$. Find the maximum and minimum values of $|m|$.

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