
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 \(\mathcal{F}\) of subsets of \(\{1,2, \ldots, 2020\}\), known to both players. Then, Calvin and Hobbes take turns choosing a number from \(\{1,2, \ldots, 2020\}\) 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 \(\mathcal{F}\), otherwise Hobbes wins. What is the largest possible size of a family \(\mathcal{F}\) that Hobbes could pick while still having a winning strategy?

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