
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
A student drew a $50 \times 50$ empty table and wrote a number above each column and to the left of each row. It turned out that all 100 numbers written are different, with 50 of them being rational and 50 being irrational. Then, in each cell of the table, he wrote the product of the numbers written next to its row and column (like in a 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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