
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
Alice rolls a fair eight-sided die each morning. If Alice rolls a number that is a perfect square, she has toast with jam. If she rolls a prime number, she has toast with butter. If she rolls any other number, she skips breakfast. In a leap year, what is the expected value of the difference between the number of days Alice has toast with butter and the number of days she has toast with jam?

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