
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 above each column and to the left of each row. It turned 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 sum of the numbers written next to its row and its column ("addition table"). What is the maximum number of sums in this table that could turn out to be rational numbers?

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