
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
If \( a \geq b \geq c \), \( a + b + c = 0 \), and \( x_{1} \), \( x_{2} \) are the two real roots of the quadratic equation \( a x^{2} + b x + c = 0 \), then the sum of the maximum and minimum values of \( \left| x_{1}^{2} - x_{2}^{2} \right| \) is \(\quad\) .

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