
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
Marisa has a collection of \(2^{8}-1=255\) distinct nonempty subsets of \(\{1,2,3,4,5,6,7,8\}\). For each step, she takes two subsets chosen uniformly at random from the collection, and replaces them with either their union or their intersection, chosen randomly with equal probability. (The collection is allowed to contain repeated sets.) She repeats this process \(2^{8}-2=254\) times until there is only one set left in the collection. What is the expected size of this set?

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