KöMaL 2011 C1064

The real numbers $x$, $y$, $z$, in this order, are the first three terms of a non-constant arithmetic progression. Given that cos $x$+cos $y$+cos $z$=1 and $\sin x + \sin y + \sin z = \frac{1}{\sqrt 2}$, find the tangent of the 12th term.