Diophantine equations and the number 4

Suppose x2+y2+z2+1=4xyz, x^2 + y^2 + z^2 + 1 = 4xyz, where x,y,z x,y,z are integers satisfying the inequalities 1<xyz 1 < x \le y \le z . Find the minimum possible value of z z .

(Extra credit: prove that the equation x2+y2+z2+1=mxyz x^2 + y^2 + z^2 + 1 = mxyz has no integer solutions if m m is a positive integer not equal to 44.)

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