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

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