
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
Set  $X$  is called *fancy* if it satisfies all of the following conditions: 


- The number of elements of  $X$  is  $2022$ .
- Each element of  $X$  is a closed interval contained in  $[0, 1]$ .
- For any real number  $r \in [0, 1]$ , the number of elements of  $X$  containing  $r$  is less than or equal to  $1011$ .

For *fancy* sets  $A, B$ , and intervals  $I \in A, J \in B$ , denote by  $n(A, B)$  the number of pairs  $(I, J)$  such that  $I \cap J \neq \emptyset$ . Determine the maximum value of  $n(A, B)$ . 

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