8.3. For all triples $(x, y, z)$ satisfying the system

$$
\left\{\begin{array}{l}
2 \cos x=\operatorname{ctg} y \\
2 \sin y=\operatorname{tg} z \\
\cos z=\operatorname{ctg} x
\end{array}\right.
$$

find the smallest value of the expression $\sin x+\cos z$.
Answer. $-\frac{5 \sqrt{3}}{6} \approx-1.44$.