
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

In each cell of a $100 \times 100$ square, a certain natural number is written. We call a rectangle, whose sides follow the grid lines, good if the sum of the numbers in all its cells is divisible by 17. It is allowed to paint all the cells in some good rectangle simultaneously. It is prohibited to paint any cell twice. What is the greatest $d$ for which it is possible to paint at least $d$ cells regardless of the arrangement of numbers?

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