
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
On February 28, a doctor prescribed pills to Absent-Minded Mathematician for absent-mindedness. He has to take one pill a day after lunch. The Mathematician bought two bottles, each with 10 pills. Each day, starting from March 1, the Mathematician takes one bottle to work (randomly choosing one), and after lunch, he takes the pill from the bottle.

a) Find the probability that on March 14, the Mathematician will find an empty bottle in his pocket for the first time.

b) Find the expected number of pills the Mathematician has taken by the time he discovers that the bottle is empty.

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