
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
Eight balls are arranged in a circle. Chris chooses two balls either adjacent or exactly two balls apart at random and interchanges them. He has a $\frac{2}{3}$ likelihood of choosing adjacent balls and a $\frac{1}{3}$ likelihood of choosing balls two apart. Silva then chooses under the same conditions but independently of Chris. What is the expected number of balls that return to their original positions after both Chris’s and Silva’s transpositions?
A) 1.8
B) 2.1
C) 2.5
D) 2.7
E) 3.0

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