
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
The numbers  $1, 2, \ldots, 2012$  are written on a blackboard. Each minute, a student goes up to the board, chooses two numbers  $x$  and  $y$ , erases them, and writes the number  $2x+2y$  on the board. This continues until only one number  $N$  remains. Find the remainder when the maximum possible value of  $N$  is divided by 1000.

*Victor Wang.*

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