
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 the side lengths of triangle \( \triangle ABC \) are \( a \), \( b \), and \( c \), and they satisfy the equation:
\[ abc = 2(a-1)(b-1)(c-1). \]

Determine if there exists a triangle \( \triangle ABC \) with all side lengths being integers. If such a triangle exists, find the side lengths; if not, provide a reason.

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