
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
Calvin and Hobbes play a game. First, Hobbes picks a family  $F$  of subsets of  $\{1, 2, . . . , 2020\}$ , known to both players. Then, Calvin and Hobbes take turns choosing a number from  $\{1, 2, . . . , 2020\}$  which is not already chosen, with Calvin going first, until all numbers are taken (i.e., each player has  $1010$  numbers). Calvin wins if he has chosen all the elements of some member of  $F$ , otherwise Hobbes wins. What is the largest possible size of a family  $F$  that Hobbes could pick while still having a winning strategy?


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