I don't think that they store the exact cards that are pulled. That would be crazy and easily disproven.
I think when they generate the board after a clear, the position on the board that will trigger the board reset is identified.
Now to the random generation factor. This could depend on how the random generation is coded and what steps they are taking to make sure that the random board reset trigger is not in the same position as the two special card triggers. Also we don't know if there are any other rules around where the board reset card can be. Random isn't always as random as expected when it has to pass validations.

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