I challenge you to solve this properly!

Algebra Level pending

\(a,b,c\) are the cube roots of \(p, (p<0)\) then for any permissible values of \(x,y,z\) which is given by

\(\left| \frac { xa+yb+zc }{ xb+yc+za } \right| +({ a }_{ 1 }^{ 2 }-2{ b }_{ 1 }^{ 2 })\omega +([x]+[y]+[z]){ \omega }^{ 2 }=0\)

(where \(\omega\) is a cube root of unity, \({ a }_{ 1 }\) is a positive real and \({ b }_{ 1 }\) is a prime number). Find the value of \(\left[ x+{ a }_{ 1 } \right] +\left[ y+b_{ 1 } \right] +\left[ z \right] \) where \([.] denotes GIF\).

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