More fun in 2016, Part 4

Find the largest integer nn such that n,n+2016n, n+2016, and n2016n-2016 are all perfect squares.

Enter 666 if you come to the conclusion that no such nn exists.

Bonus Question (very hard): Does there exist a rational number qq such that q,q+2015q,q+2015 and q2015q-2015 are all squares of rational numbers?

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