
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 \( A = \{1, 2, \cdots, 2012\} \),
\[ B = \{1, 2, \cdots, 19\}, \]

and let \( S \) be the set of all subsets of \( A \). Find the number of functions \( f: S \rightarrow B \) such that
\[ 
f\left(A_{1} \cap A_{2}\right)=\min \left\{f\left(A_{1}\right), f\left(A_{2}\right)\right\}
\]
holds for any \( A_{1}, A_{2} \in S \).

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