
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
Let  $ N$  be the number of ordered pairs of nonempty sets  $ \mathcal{A}$  and  $ \mathcal{B}$  that have the following properties:

•  $ \mathcal{A} \cup \mathcal{B} \equal{} \{1,2,3,4,5,6,7,8,9,10,11,12\}$ ,
•  $ \mathcal{A} \cap \mathcal{B} \equal{} \emptyset$ ,
• The number of elements of  $ \mathcal{A}$  is not an element of  $ \mathcal{A}$ ,
• The number of elements of  $ \mathcal{B}$  is not an element of  $ \mathcal{B}$ .

Find  $ N$ .

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