
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 three quadratic polynomials \( f(x)=a x^{2}+b x+c \), \( g(x)=b x^{2}+c x+a \), and \( h(x)=c x^{2}+a x+b \), where \( a, b, c \) are distinct non-zero real numbers. From these, three equations \( f(x)=g(x) \), \( f(x)=h(x) \), and \( g(x)=h(x) \) are formed. Find the product of all roots of these three equations, assuming each equation has two distinct roots.

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