
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
Each day, cat Basilio and fox Alisa visit all 20 courtyards in the capital of Fool's Country. In each courtyard, they are either given or not given one gold coin with a probability of $\frac{1}{2}$. By the end of the day, if the number of received gold coins is even, then the fox and the cat share them equally. If the number of coins is odd, they share all the coins except one equally, and cat Basilio keeps the last coin as the commercial director of the enterprise. Find the expected number of coins received by the cat per day.

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