
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
Determine the greatest positive integer  $A{}$  with the following property: however we place the numbers  $1,2,\ldots, 100$  on a  $10\times 10$  board, each number appearing exactly once, we can find two numbers on the same row or column which differ by at least  $A{}$ .

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