
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  $P$  be the set of all  $2012$  tuples  $(x_1, x_2, \dots, x_{2012})$ , where  $x_i \in \{1,2,\dots 20\}$  for each  $1\leq i \leq 2012$ . The set  $A \subset P$  is said to be decreasing if for each  $(x_1,x_2,\dots ,x_{2012} ) \in A$  any  $(y_1,y_2,\dots, y_{2012})$  satisfying  $y_i \leq x_i (1\leq i \leq 2012)$  also belongs to  $A$ . The set  $B \subset P$  is said to be increasing if for each  $(x_1,x_2,\dots ,x_{2012} ) \in B$  any  $(y_1,y_2,\dots, y_{2012})$  satisfying  $y_i \geq x_i (1\leq i \leq 2012)$  also belongs to  $B$ . Find the maximum possible value of  $f(A,B)= \dfrac {|A\cap B|}{|A|\cdot |B|}$ , where  $A$  and  $B$  are nonempty decreasing and increasing sets ( $\mid \cdot \mid$  denotes the number of elements of the set).

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