A strange equation V

Algebra Level 5

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.

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