
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
Let \( f \) be a quadratic function of \( x \). If \( 2y \) is a root of \( f(x-y) \), and \( 3y \) is a root of \( f(x+y) \), what is the product of the roots of \( f(x) \)?
(a) \( 6y^{2} \)  
(b) \( 5y^{2} \)  
(c) \( 4y^{2} \)  
(d) \( 3y^{2} \)

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