
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 -10, -9, -8, ..., 9, 10 are arranged in a line. A player places a token on 0 and throws a fair coin 10 times. For each head, the token is moved one place to the left and for each tail, it is moved one place to the right. If we color one or more numbers black and the remainder white, we find that the chance of the token ending up on a black number is m/n with m + n = 2001. What is the largest possible total for the black numbers?

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