
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
A hotel packs breakfast for each of three guests. Each breakfast should now consist of four types of rolls: one nut, one cheese, one fruit, and one seed roll. The preparer wrapped each of the 12 rolls and, once wrapped, the rolls were indistinguishable from one another. She randomly put four rolls in a bag for each of the guests. Given that the probability each guest got one roll of each type is $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m+n$.

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