
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
Given the sets $\mathrm{A}=\{1,2, \ldots, 2012\}$ and $\mathrm{B}=\{1,2, \ldots, 19\}$, let $\mathrm{S}$ be the set of all subsets of $\mathrm{A}$. Determine the number of functions $\mathrm{f}: \mathrm{S} \rightarrow \mathrm{B}$ such that for all $\mathrm{A}_{1} , \mathrm{A}_{2} \in \mathrm{S}$, it holds that $\mathrm{f}(\mathrm{A}_{1} \cap \mathrm{A}_{2}) = \min \{\mathrm{f}(\mathrm{A}_{1}), \mathrm{f}(\mathrm{A}_{2})\}$.

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