
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
Two wise men play the following game. The numbers $0,1,2, \ldots, 1024$ are written down. The first wise man crosses out 512 numbers (of his choice), the second wise man then crosses out 256 of the remaining numbers, and then the first wise man crosses out 128 numbers, and so on. On the tenth turn, the second wise man crosses out one number; two numbers remain. After this, the second wise man pays the first wise man the difference between these two numbers. How should the first wise man play to his advantage? How should the second? How much will the second wise man pay the first if both play optimally?

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