Newton's Sums with pi?

Algebra Level 5

x+y+z=1x2+y2+z2=πx3+y3+z3=π2+3z23z+1x+y+z=1\\ { x }^{ 2 }+{ y }^{ 2 }+{ z }^{ 2 }=\pi\\ { x }^{ 3 }+{ y }^{ 3 }+{ z }^{ 3 }={ \pi }^{ 2 }+3{ z }^{ 2 }-3z+1\\

This set of equations is true for three complex numbers x,y,zx, y, z.

If xyzxy=ABπCxyz-xy=\frac { A }{ B } {\pi }^{ C } for positive integers A,B,CA, B, C and gcd(A,B)=1\gcd(A,B)=1, then find A+B+CA+B+C.

First do this.

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