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Distinct Integer Sum Count

Given that $ x $, $ \frac{1}{x} $, $ y $, $ \frac{1}{y} $, $ z $ and $ \frac{1}{z} $ are all integers, how many distinct values of $ x+ y+ z $ are possible?

1
2
3
+
4
5
6
-
7
8
9
$\frac{a}{b}$
.
0
=
%

$a^n$
$a^{\circ}$
$a_n$
$\sqrt{}$
$\sqrt[n]{}$
$\pi$
$\ln{}$
$\log$
$\theta$
$\sin{}$
$\cos{}$
$\tan{}$

$($
$)$
$[$
$]$
$\cap$
$\cup$
$,$
$\infty$