
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
Oleg drew an empty $50 \times 50$ table and wrote a number on top of each column and to the left of each row. It turns out that all 100 written numbers are different, with 50 of them being rational and the remaining 50 being irrational. Then, in each cell of the table, he wrote the product of the numbers written next to its row and its column ("multiplication table"). What is the maximum number of products in this table that could be rational numbers?

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