Diophantine equations with three variables

Algebra Level pending

The equation \(29x + 30y + 31z = 366\) has \(n\) solutions \((x,y,z)\) where \(x,y,z\) are positive (not necessarily distinct) integers.

If \((X,Y,Z)\) is the sole solution in which \(X,Y,Z\) are all distinct, then find \(n + X + Y + Z\).

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