
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
Ten delegates, three from three different countries and one independent delegate, randomly select chairs at a round table that seats ten people. Calculate the probability that each delegate sits next to at least one delegate from another country or the independent delegate. Express the probability in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, and find $m+n$.

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