
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
In a game, the numbers $1, 2, \cdots, 2004$ are initially written on the board. Each step in the game involves the following:
1) Selecting a subset of the numbers on the board.
2) Writing the sum of these selected numbers modulo 11 on the board.
3) Erasing the originally selected numbers from the board.
When the game proceeds such that only two numbers are left on the board, one of which is 1000, what is the other number?

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