
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 that \(a, b, c\) are positive integers such that the roots of the three quadratic equations
\[
x^2 - 2ax + b = 0, \quad x^2 - 2bx + c = 0, \quad x^2 - 2cx + a = 0
\]
are all positive integers. Determine the maximum value of the product \(abc\).

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