
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
1. Player A writes down two rows of 10 positive integers, one under the other. The numbers must be chosen so that if \(a\) is under \(b\) and \(c\) is under \(d\), then \(a + d = b + c\). Player B is allowed to ask for the identity of the number in row \(i\), column \(j\). How many questions must he ask to be sure of determining all the numbers?

2. An \(m \times n\) array of positive integers is written on the blackboard. It has the property that for any four numbers \(a, b, c, d\) with \(a\) and \(b\) in the same row, \(c\) and \(d\) in the same row, \(a\) above \(c\) (in the same column) and \(b\) above \(d\) we have \(a + d = b + c\). If some numbers are wiped off, how many must be left for the table to be accurately restored?

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