
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  $A$  be a set contains  $2000$  distinct integers and  $B$  be a set contains  $2016$  distinct integers.  $K$  is the numbers of pairs  $(m,n)$  satisfying \[ \begin{cases} m\in A, n\in B |m-n|\leq 1000 \end{cases} \] Find the maximum value of  $K$ 

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