Minimum value of an expression in two variables $$\sqrt{x^2-7\sqrt{2}x+49} + \sqrt{x^2+y^2-\sqrt{2}xy} + \sqrt{50+y^2-10y}$$
$x$ and $y$ are positive real numbers, what is the minimum value?
I have tried finding the minimum value of the first expression and the third expression, with the first expression being $\frac{7}{\sqrt{2}}$ while the second expression being $5$, I do not know how to move on from this (note that the answer is an integer from $0$ to $999$).

let $$f(x,y)=\sqrt{x^2-7\sqrt{2}x+49}+\sqrt{x^2+y^2-\sqrt{2}xy}+\sqrt{50+y^2-10y}$$
then solve
$$f_x=1/2\,{\frac {2\,x-7\,\sqrt {2}}{\sqrt {{x}^{2}-7\,\sqrt {2}x+49}}}+1/2
\,{\frac {2\,x-\sqrt {2}y}{\sqrt {{x}^{2}+{y}^{2}-\sqrt {2}xy}}}
=0$$
and
$$f_y=1/2\,{\frac {2\,y-\sqrt {2}x}{\sqrt {{x}^{2}+{y}^{2}-\sqrt {2}xy}}}+1/
2\,{\frac {2\,y-10}{\sqrt {{y}^{2}-10\,y+50}}}
=0$$
