PAMO 2013 Problem 6

Let $x$ , $y$ , and $z$ be real numbers such that $x<y<z<6$ . Solve the system of inequalities: $$\left\{\begin{array}{cc} \dfrac{1}{y-x}+\dfrac{1}{z-y}\le 2 \dfrac{1}{6-z}+2\le x \end{array}\right.$$