
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

There are 20 counterfeit bills on which seven-digit serial numbers need to be inscribed. Each bill has 7 empty slots for digits. Basilio the cat names one digit at a time, choosing between "1" and "2" (these are the only digits he knows), and Alice writes the named digit in any empty slot on any bill and shows the result to Basilio. Once all the slots are filled, Basilio takes as many bills with unique serial numbers as possible (if there are multiple bills with the same number, he only takes one). The rest of the bills are taken by Alice. What is the maximum number of bills that Basilio can get, no matter how Alice acts?

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