
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
Suppose  $ A_1,\dots, A_6$  are six sets each with four elements and  $ B_1,\dots,B_n$  are  $ n$  sets each with two elements, Let  $ S \equal{} A_1 \cup A_2 \cup \cdots \cup A_6 \equal{} B_1 \cup \cdots \cup B_n$ . Given that each elements of  $ S$  belogs to exactly four of the  $ A$ 's and to exactly three of the  $ B$ 's, find  $ n$ .

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