
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  $f(x,y)$  and  $g(x,y)$  be real valued functions defined for every  $x,y \in \{1,2,..,2000\}$ . If there exist  $X,Y \subset \{1,2,..,2000\}$  such that  $s(X)=s(Y)=1000$  and  $x\notin X$  and  $y\notin Y$  implies that  $f(x,y)=g(x,y)$  than, what is the maximum number of  $(x,y)$  couples where  $f(x,y)\neq g(x,y)$ .

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