Pythagorean Generator Matrix

(xyyyxyyyz)(abc)=(abc)\begin{pmatrix}x & y & y \\ y & x & y \\ y & y & z \end{pmatrix}\begin{pmatrix}a \\ b \\ c \end{pmatrix} = \begin{pmatrix}a' \\ b' \\ c' \end{pmatrix} If the above equation holds true, then there exist coprime positive integers x,y,zx, y, z such that any Pythagorean triple (a,b,c)(a, b, c) produces another Pythagorean triple (a,b,c).\big(a', b', c'\big).

Find the sum of such x,y,x, y, and z.z.

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