
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 polynomials \( f_{1}(x) = x^{2} - a x - 3 \), \( f_{2}(x) = x^{2} + 2 x - b \), \( f_{3}(x) = 3 x^{2} + (2 - 2 a) x - 6 - b \), and \( f_{4}(x) = 3 x^{2} + (4 - a) x - 3 - 2 b \). Let the differences of their roots be denoted as \( A, B, C \), and \( D \) respectively. It is known that \( |C| \neq |D| \). Find the ratio \( \frac{A^{2} - B^{2}}{C^{2} - D^{2}} \). The values of \( A, B, C, D, a, \) and \( b \) are not given.

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