
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 adults enter a room, remove their shoes, and toss their shoes into a pile. Later, a child randomly pairs each left shoe with a right shoe without regard to which shoes belong together. The probability that for every positive integer \( k<6 \), no collection of \( k \) pairs made by the child contains the shoes from exactly \( k \) of the adults is expressed as \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers. Find \( m+n \).

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