
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  $R$  be the set of points  $(x, y)$  such that  $x$  and  $y$  are positive,  $x + y$  is at most 2013, and
\[
  \lceil x \rceil \lfloor y \rfloor = \lfloor x \rfloor \lceil y \rceil.
\]
Compute the area of set  $R$ .  Recall that  $\lfloor a \rfloor$  is the greatest integer that is less than or equal to  $a$ , and  $\lceil a \rceil$  is the least integer that is greater than or equal to  $a$ .

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