Direct Proof · Challenge problem

The identity $(a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2$ shows that a product of sums of two squares is again one. In how many ways can $1105 = 5\cdot13\cdot17$ be written as $x^2 + y^2$ with integers $0 < x < y$?