Diophantine equations and the number 4

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

(Extra credit: prove that the equation \( x^2 + y^2 + z^2 + 1 = mxyz \) has no integer solutions if \( m \) is a positive integer not equal to \(4\).)

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