
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  $A=\{1,2,\ldots,2012\}, \: B=\{1,2,\ldots,19\}$  and  $S$  be the set of all subsets of  $A.$  Find the number of functions  $f : S\to B$  satisfying  $f(A_1\cap A_2)=\min\{f(A_1),f(A_2)\}$  for all  $A_1, A_2 \in S.$ 

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