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With the circumcircle's radius $r_c$ , we can see that the hypotenuse $c$ of the triangle is $2 \times r_c = 4034$ .

Now, the radius $r_i$ of the incircle of a right triangle with sides $a, b$ and hypotenuse $c$ is given by $\displaystyle r_i = \frac{a + b - c}{2}$ :

$\displaystyle \begin{aligned} r_i & = \frac{a + b - c}{2} \\ \implies a + b & = 2r_i + c \\ & = 2 \cdot 217 + 4034 \\ & = \boxed{4468} \end{aligned}$

Using Thales' Theorem we get that $2017=\dfrac{x+y}{2}$ . Notice that $r=217$ . From that

$a+b=x+y+2r=2017*2+217*2=\boxed{4468}$