
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
Angela, Bill, and Charles each independently and randomly choose a subset of  $\{ 1,2,3,4,5,6,7,8 \}$  that consists of consecutive integers (two people can select the same subset). The expected number of elements in the intersection of the three chosen sets is  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .

*Proposed by **Th3Numb3rThr33***

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