
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
Consider the set  $A = \{0, 1, 2, \dots , 9 \}$  and let  $(B_1,B_2, \dots , B_k)$  be a collection of nonempty subsets of  $A$  such that  $B_i \cap B_j$  has at most two elements for  $i \neq j$ . What is the maximal value of  $k \ ?$ 

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