
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
Twelve delegates, three each from four different countries, randomly select chairs at a round table that seats twelve people. Let the probability that each delegate sits next to at least one delegate from another country be expressed as a reduced fraction \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n\).

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