
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 $2 \times 2019$ grid is filled with real numbers such that the 2019 numbers in the top row are all distinct, and the bottom row contains the same 2019 numbers in a different order. In each of the 2019 columns, the two different numbers sum to a rational number. What is the maximum number of irrational numbers that can appear in the first row?

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