
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
8.6 Person A and Person B are playing a math game. They start with the numbers \(0, 1, 2, \ldots, 1024\). Person A crosses out 512 numbers, then Person B crosses out 256 of the remaining numbers, then Person A crosses out 128, and so on. In the 10th step, Person B crosses out 1 number, leaving two numbers remaining. At this point, Person B needs to pay Person A an amount equal to the difference between the remaining two numbers. What strategy should each player follow to maximize their advantage? If both players play optimally, how much will Person B pay Person A?

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