
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
Kevin has  $255$  cookies, each labeled with a unique nonempty subset of  $\{1,2,3,4,5,6,7,8\}$ . Each day, he chooses one cookie uniformly at random out of the cookies not yet eaten. Then, he eats that cookie, and all remaining cookies that are labeled with a subset of that cookie (for example, if he chooses the cookie labeled with  $\{1,2\}$ , he eats that cookie as well as the cookies with  $\{1\}$  and  $\{2\}$ ). The expected value of the number of days that Kevin eats a cookie before all cookies are gone can be expressed in the form  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .

*Proposed by Ray Li*

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