
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]


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Given the quadratic polynomials \(f_{1}(x)=x^{2}-2x+a\), \(f_{2}(x)=x^{2}+bx-2\), \(f_{3}(x)=4x^{2}+(b-6)x+3a-2\), and \(f_{4}(x)=4x^{2}+(3b-2)x-6+a\), let the differences of their roots be \(A\), \(B\), \(C\), 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}
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