NMTC Practice Part 9

If xx, yy, zz are distinct natural number such that x<y<zx<y<z and 1x1xy1xyz=1997\frac{1}{x}-\frac{1}{xy}-\frac{1}{xyz}=\frac{19}{97}, then find x+y+zx+y+z.

Part of the set NMTC Practice Problems.
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