Only Squares Allowed

Algebra Level 5

{x2+y2+z2=194x2z2+y2z2=4225 \begin{cases} x^2 + y^2 + z^2 &= 194\\ x^2z^2 + y^2z^2 &= 4225\\ \end{cases} How many ordered triples (x,y,z)(x,y,z) of integer solutions satisfy the above system of equations?

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