
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 \(a, b, c\) be real numbers such that
\[ 3ab + 2 = 6b, \quad 3bc + 2 = 5c, \quad 3ca + 2 = 4a. \]

Suppose the only possible values for the product \(abc\) are \(\frac{r}{s}\) and \(\frac{t}{u}\), where \(\frac{r}{s}\) and \(\frac{t}{u}\) are both fractions in lowest terms. Find \(r+s+t+u\).

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