Answer: $\left(\frac{1}{3} ; \frac{1}{3} ; \frac{1}{3}\right)$
4. For which triples of positive numbers $(x ; y ; z)$, satisfying the system $\left\{\begin{array}{l}\sin 3 x-\sin 3 y=3(x-y) \\ \cos 4 x-\cos 4 z=5(z-x)\end{array}, \quad\right.$ does the expression $x+y+z-2 \operatorname{arctg}(x+y+z) \quad$ take the smallest possible value