A conversation in school

Andrei decides to do some random computation in his mind. He starts out with the number \(x\). First, he multiplies his number by \(3\). Then, he squares his new number. Lastly, he subtracts \(8\) times the square of his original number from the new number. His final result is the units digit of \(3^{2016}+2(25^{1000})\). When Kostya tries to figure out the original number, he discovers that there are multiple answers. Andrei then tells Kostya that his starting number was his date of birth (Kostya did not know it). What number did Andrei start with?

This is adapted from a problem in the IMO.
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