
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
A writes the positive integers \(1, 2, \cdots, 2009\) on the blackboard. Then, facing away from the blackboard, A lets B erase some of these numbers and add the remainder when the sum of the erased numbers is divided by 7. After repeating this process several times until only two numbers are left on the blackboard, one of which is a single-digit number, A asks B: "What is the larger of the two remaining numbers?" The answer is "100." What is the single-digit number among the remaining two numbers?

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