
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
A confectionary factory produces $N$ types of candy. For New Year, the factory gave each of 1000 school students a gift containing candies of several types (the composition of gifts could differ). Each student noticed that for any 11 types of candy, they received at least one candy of some type from those 11 types. However, it turned out that for any two types of candy, there exists a student who received a candy of exactly one of those two types. Find the maximum possible value of $N$.

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