A strange equation V

The seven real roots of the equation $x^7+28x^6+308x^5+1680x^4+4704x^3+6272x^2+3136x-128\sqrt{2}+256 = 0$ are, in ascending order: $x_{1}=-a-b\cos\left(\dfrac{c\pi}{d}\right)$ $x_{2}=-a-b\cos\left(\dfrac{e\pi}{d}\right)$ $x_{3}=-a-b\cos\left(\dfrac{f\pi}{d}\right)$ $x_{4}=-a-b\cos\left(\dfrac{g\pi}{d}\right)$ $x_{5}=-a+b\cos\left(\dfrac{h\pi}{d}\right)$ $x_{6}=-a+j\sqrt{j}$ $x_{7}=-a+b\cos\left(\dfrac{k\pi}{d}\right)$ All the variables are positive integer numbers, and the angles are in radians between $0$ and $\frac{\pi}{2}$. Find $a+b+c+d+e+f+g+h+j+k$.

You may also try Part VI.