
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 \( f_{1}(x) = x^{2} - x + 2a \), \( f_{2}(x) = x^{2} + 2bx + 3 \), \( f_{3}(x) = 4x^{2} + (2b-3)x + 6a + 3 \), and \( f_{4}(x) = 4x^{2} + (6b-1)x + 9 + 2a \), let the differences of their roots be respectively \( A \), \( B \), \( C \), and \( D \). It is known that \( |A| \neq |B| \). Find the ratio \( \frac{C^{2} - D^{2}}{A^{2} - B^{2}} \). The values of \( A, B, C, D, a, \) and \( b \) are not given.

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